Development of predictive force models for classical
orthogonal and oblique cutting and turning
operations incorporating tool flank wear effects
By
WENGE SONG B.E. (Mech.) and M.E. (Mech.)
A thesis submitted to the Queensland University of Technology for the degree of
Doctor of Philosophy
School of Engineering Systems
Queensland University of Technology
2006
I
KEYWORDS
Orthogonal cutting, Oblique cutting, Turning operations, Flank wear, Wearland
force, Cutting force model, Mechanics of cutting, Chip formation , Equivalent
cutting edge
II
ABSTRACT
Classical orthogonal and oblique cutting are the fundamental material removal or
machining processes to which other practical machining processes can be related in
the study and modelling of the machining processes. In the last century, a large
amount of research and development work has been done to study and understand the
various machining processes with a view to improving the processes for further
economic (cost and productivity) gains. However, many aspects of the cutting
processes and cutting performance remains to be fully understood in order to
increase the cutting capability and optimize the cutting processes; in particular, there
is little study to understand the effects of the inevitable tool wear on the machining
processes.
This thesis includes an extensive literature review on the mechanics of cutting
analysis. Considerable work has been carried out in past decades on the fundamental
analysis of ‘sharp’ tool cutting. Although some work has been reported on the effects
of tool flank wear on the cutting performance, there is a general lack of the
fundamental study of the effects of the flank wear on the basic cutting or chip
formation process. It has been well documented that tool flank wear results in an
increase in the cutting forces. However, it was not known if this force increase is a
result of the change in the chip formation process, and/or the rubbing or ploughing
forces between the tool flank and the workpiece. In work carried out since the early
1980s, the effects of the so-called edge forces have been considered when the tool is
not absolutely sharp. Little has been reported to further develop fundamental cutting
theories to understand applications to more relevant the practical situation, i.e. to
consider the tool wear effects.
Based on the findings of the literature review, an experimental investigation is
presented in the first part of the thesis to study the effects of tool flank wear on the
basic cutting or chip formation process by examining the basic cutting variables and
performance in the orthogonal cutting process with tool flank wear. The effects of
tool flank wear on the basic cutting variables are discussed by a comprehensive
analysis of the experimental data. It has been found that tool flank wear does not
III
affect the basic cutting variables (i.e. shear angle, friction angle and shear stress). It
is therefore deduced that the flank wear does not affect the basic chip formation
process in the shear zone and in the tool-chip interface. The study also finds that tool
flank wear causes an increase in the total cutting forces, as can be expected and such
an increase is entirely a result of the rubbing or ploughing forces on the tool
wearland. The significance of this finding is that the well-developed machining
theories for ‘sharp’ tools can be used in modelling the machining processes when
tool flank wear is present, rather than study the machining process and develop
machining theories from scratch. The ploughing forces can be modelled for
incorporation into the overall cutting force prediction.
The experimental study also allows for the forces on the wearland (or wearland
force) and edge forces to be separated from the total measured forces. The wearland
force and edge force models are developed in empirical form for force prediction
purpose. In addition, a database for the basic cutting variables or quantities is
established for use in modelling the cutting forces. The orthogonal cutting force
model allowing for the effects of flank wear is developed and verified by the
experimental data.
A comprehensive analysis of the mechanics of cutting in the oblique cutting process
is then carried out. Based on this analysis, predictive cutting force models for oblique
cutting allowing for the effects of flank wear are proposed. The wearland force and
edge force are re-considered by analysing the oblique cutting process and the
geometrical relation. The predictive force models are qualitatively and quantitatively
assessed by oblique cutting tests. It shows that the model predictions are in excellent
agreement with the experimental data.
The modelling approach is then used to develop the cutting force models for a more
general machining process, turning operation. By using the concept of an equivalent
cutting edge, the tool nose radius is allowed for under both orthogonal and oblique
cutting conditions. The wearland forces and edge forces are taken into consideration
by the integration of elemental forces on the tool flank and the cutting edge,
respectively. The cutting forces in turning operations are successfully predicted by
using the basic cutting quantity database established in the orthogonal cutting
IV
analysis. The models are verified by turning operation tests. It shows that the model
predictions are in excellent agreement with the experimental results both
qualitatively and quantitatively.
The major findings, research impacts and practical implications of the research are
finally highlighted in the conclusion. The modelling approach considering the flank
wear effects in the classical orthogonal and oblique cutting and turning operations
can be readily extended to other machining operations, such as drilling and milling.
V
TABLE OF CONTENTS
KEYWORDS………....................................................................................................I
ABSTRACT……........................................................................................................ II
TABLE OF CONTENTS........................................................................................... V
LIST OF FIGURES ..............................................................................................VIII
LIST OF TABLES .................................................................................................... X
LIST OF TABLES .................................................................................................... X
STATEMENT OF ORIGINAL AUTHORSHIP ..................................................XI
ACKNOWLEDGMENTS ..................................................................................... XII
NOMENCLATURE..............................................................................................XIII
CHAPTER 1 INTRODUCTION.......................................................................... 1
1.1 BACKGROUND ....................................................................................... 1 1.2 OBJECTIVES OF THE RESEARCH............................................................. 3 1.3 THESIS ORGANISATION ......................................................................... 4
CHAPTER 2 LITERATURE REVIEW.............................................................. 5
2.1 ORTHOGONAL AND OBLIQUE CUTTING MODELS FOR PERFECT SHARP TOOLS.................................................................................................... 7 2.1.1 Orthogonal cutting models for sharp tools............................... 7 2.1.2 Oblique cutting models for sharp tools .................................. 13
2.2 ORTHOGONAL AND OBLIQUE CUTTING MODELS INCORPORATING EDGE FORCE.................................................................................................. 15 2.2.1 Orthogonal cutting models incorporating edge force............. 16 2.2.2 Oblique cutting models incorporating edge force.................. 17
2.3 PREDICTIVE FORCE MODELS FOR TURNING OPERATIONS ..................... 19 2.3.1 Cutting force models for single straight cutting edge ............ 19 2.3.2 Mechanics of cutting analysis for turning operations with
sharp cornered and nose radius edged tools........................... 21 2.4 CUTTING TOOL WEAR AND ITS EFFECTS ON THE CUTTING FORCES ....... 26
2.4.1 Wear on cutting tool in machining......................................... 27 2.4.2 Effects of tool flank wear on the cutting forces ..................... 29
2.5 CONCLUDING REMARKS ...................................................................... 40
CHAPTER 3 MECHANICS OF CUTTING ANALYSIS FOR ORTHOGONAL CUTTING ALLOWING FOR TOOL FLANK WEAR EFFECTS ........................................................................................................ 43
3.1 ORTHOGONAL CUTTING TESTS ............................................................ 45 3.2 DATA PROCESSING AND EVALUATION PROCEDURE.............................. 49
3.2.1 Methodologies of data analysis.............................................. 52 3.2.2 Shear angle relationship and basic cutting quantities database.. ................................................................................................ 54
3.3 THE CHARACTERISTICS OF THE MEASURED CUTTING FORCES .............. 55 3.4 THE BASIC CUTTING QUANTITIES IN THE SHEAR ZONE AND TOOL CHIP
INTERFACE .......................................................................................... 64 3.4.1 The characteristics of chip length ratio .................................. 65
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3.4.2 The characteristics of shear stress .......................................... 70 3.4.3 The characteristics of friction angle....................................... 74 3.4.4 Shear angle relationship and characteristics .......................... 78 3.4.5 Summary ................................................................................ 79
3.5 ANALYSIS OF THE RUBBING FORCE COMPONENTS ............................... 80 3.6 PROPOSED ORTHOGONAL CUTTING FORCE MODELS WITH TOOL FLANK
WEAR .................................................................................................. 84 3.6.1 Model formulation ................................................................. 84 3.6.2 Basic cutting quantities database ........................................... 86 3.6.3 Assessment of the predictive model....................................... 88
3.7 CONCLUDING REMARKS ...................................................................... 91
CHAPTER 4 PREDICTIVE CUTTING FORCE MODELS FOR OBLIQUE CUTTING ALLOWING FOR TOOL FLANK WEAR EFFECTS ........................................................................................................ 94
4.1 FORMATION OF THE PREDICTIVE CUTTING FORCE MODEL.................... 95 4.1.1 Forces required for basic chip formation ............................... 96 4.1.2 Edge force component............................................................ 98 4.1.3 Wearland force component .................................................... 99 4.1.4 Overall cutting force models for oblique cutting with tool
flank wear............................................................................. 101 4.2 MODEL VERIFICATION....................................................................... 102
4.2.1 Experimental work............................................................... 102 4.2.2 Results and discussion ......................................................... 104
4.3 CONCLUDING REMARKS .................................................................... 114
CHAPTER 5 PREDICTIVE CUTTING FORCE MODELS FOR TURNING OPERATIONS ALLOWING FOR THE EFFECTS OF FLANK WEAR ............................................................................................. 116
5.1 OVERVIEW OF THE MODELLING APPROACH ....................................... 116 5.2 FORCES FOR BASIC CHIP FORMATION IN TURNING OPERATION........... 120
5.2.1 The equivalent cutting edge concept.................................... 120 5.2.2 The chip flow angle caused by tool nose radius and the
equivalent cutting edge ........................................................ 122 5.2.3 Equivalent tool angles for the equivalent cutting edge ........ 128 5.2.4 The forces for basic chip formation ..................................... 129
5.3 EDGE FORCES IN TURNING OPERATIONS ............................................ 131 5.4 WEARLAND FORCES IN TURNING OPERATIONS .................................. 136 5.5 FINAL CUTTING FORCE MODEL FOR TURNING OPERATIONS ALLOWING
FOR TOOL FLANK WEAR ..................................................................... 140 5.6 MODEL VERIFICATION....................................................................... 142
5.6.1 Experimental lay-out and procedures................................... 143 5.6.2 Results and discussion ......................................................... 145
5.7 CONCLUDING REMARKS .................................................................... 151
CHAPTER 6 FINAL CONCLUSIONS AND FUTURE WORK.................. 154
6.1 FINAL CONCLUSIONS ......................................................................... 154 6.2 MAJOR SCIENTIFIC CONTRIBUTIONS OF THE RESEARCH ................... 156 6.3 PROPOSALS FOR FURTHER STUDIES ................................................... 157
REFERENCES....................................................................................................... 158
VII
Appendix A: Measured forces, Chip length and weight in orthogonal cutting.A1 Appendix B: Measured forces in oblique cutting……………………………….A5 Appendix C: Measured forces in turning operations……………...……………A8
VIII
LIST OF FIGURES
Figure 2.1 Model of chip formation in orthogonal cutting for a ‘sharp’ tool. ............. 8 Figure 2.2 Concept of equivalent cutting edge .......................................................... 23 Figure 2.3 Types of wear on turning tools ................................................................. 28 Figure 2.4 Slipline fields: (a) wave model; (b) wave removal model; (c) chip
formation model ....................................................................................... 32 Figure 3.1 CNC lathe, workpiece and tool set-up in orthogonal cutting tests ........... 46 Figure 3.2 Diagram of cutting force measuring system............................................. 48 Figure 3.3 Data sampling at steady state area ............................................................ 49 Figure 3.4 Effect of cut thickness on measured forces. ............................................. 56 Figure 3.5 Effect of rake angle on the measured forces............................................. 56 Figure 3.6 Effect of cutting speed on the measured forces ........................................ 57 Figure 3.7 Effect of wearland size on the measured forces ....................................... 57 Figure 3.8 Effect of cut thickness on chip length ratio under different wearland sizes.
.................................................................................................................. 65 Figure 3.9 Effect of cutting speed on chip length ratio under different rake angles.. 68 Figure 3.10 Shear angle vs cutting speed................................................................... 68 Figure 3.11 Effect of tool flank wear on chip length ratio and shear angle............... 69 Figure 3.12 Effect of cut thickness on shear stress under different rake angles. ....... 70 Figure 3.13 Effect of cutting speed on shear stress under different rake angles........ 73 Figure 3.14 Effect of cut thickness on shear stress under different wearland sizes... 74 Figure 3.15 Friction angle against cut thickness under different rake angles............ 75 Figure 3.16 Friction angle vs cut speed under different rake angles. ........................ 76 Figure 3.17 Effect of wearland size on friction angle................................................ 77 Figure 3.18 Shear angle φ against (β-γ) for different wearland sizes. ....................... 78 Figure 3.19 Rubbing force components vs wearland size.......................................... 81 Figure 3.20 Orthogonal cutting model allowing for the effect of tool flank wear..... 85 Figure 3.21 Predicted and experimental cutting forces.............................................. 89 Figure 3.22 Histograms for percentage deviations between predicted and
experimental force components under different wearland sizes. ............. 91 Figure 3.23 Histograms for percentage deviations between predicted and
experimental force components for all wearland sizes. ........................... 91 Figure 4.1 Forces in oblique cutting .......................................................................... 96 Figure 4.2 Edge forces on the cutting edge in oblique cutting................................... 98 Figure 4.3 The contact area of weraland and workpiece in the oblique cutting. ....... 99 Figure 4.4 Wearland forces in oblique cutting......................................................... 100 Figure 4.5 Procedure for cutting force prediction in oblique cutting....................... 102 Figure 4.6 Predicted and experimental forces vs. cut thickness. ............................. 105 Figure 4.7 Predicted and experimental forces vs. wearland size. ............................ 105 Figure 4.8 Predicted and measured forces for different normal rake angle. ............ 106 Figure 4.9 Predicted and measured forces vs cut thickness for different speeds..... 107 Figure 4.10 Predicted and measured forces for inclination angles. ......................... 107 Figure 4.11 Percentage deviations between predicted and experimental power force
components for different VB ................................................................. 110 Figure 4.12 Percentage deviations between predicted and experimental thrust force
components for different VB ................................................................. 111 Figure 4.13 Percentage deviations between predicted and experimental radial force
components for different VB ................................................................. 112
IX
Figure 4.14 Percentage deviations between predicted and experimental cutting forces. ..................................................................................................... 113
Figure 5.1 Single-point tool angles .......................................................................... 117 Figure 5.2 Forces in general turning operation. ....................................................... 118 Figure 5.3 Elemental forces in a general turning operation. .................................... 119 Figure 5.4 Equivalent cutting edge for nose radius tool .......................................... 122 Figure 5.5 General chip flow model on rake face.................................................... 123 Figure 5.6 Geometrical relation of chip flow (Case 1). ........................................... 125 Figure 5.7 Geometrical relations of chip flow (Case 2)........................................... 126 Figure 5.8 Edge force analysis in turning operations............................................... 131 Figure 5.9 Edge forces at nose radius in turning operations. ................................... 135 Figure 5.10 Elemental wearland forces in turning operations. ................................ 137 Figure 5.11 Procedures for cutting force prediction for turning operations with tool
flank wear............................................................................................... 142 Figure 5.12 Predicted and measured forces vs feed rate f........................................ 145 Figure 5.13 Predicted and measured forces vs depth of cut d.................................. 146 Figure 5.14 Predicted and measured forces vs normal rake angle γn. ...................... 147 Figure 5.15 Predicted and measured forces vs inclination angle i. .......................... 147 Figure 5.16 Predicted and measured forces vs wearland size VB. .......................... 148 Figure 5.17 Predicted and measured forces vs feed rate for different approach angle
Cs. ........................................................................................................... 149 Figure 5.18 Percentage deviations of predicted forces from the experimental forces.
................................................................................................................ 150
X
LIST OF TABLES Table 3.1 Chemical compositions and mechanical properties of CS1020 steel ........ 45 Table 3.2 Parameters and levels used in orthogonal cutting tests.............................. 47 Table 3.3 Measured forces analysis plan ................................................................... 53 Table 3.4 Chip length ratios and shear angles analysis plan...................................... 54 Table 3.5 Shear stress analysis plan........................................................................... 54 Table 3.6 Friction angle analysis plan ....................................................................... 54 Table 3.7 Regression and correlation analysis of the measured power force
component Fcm in terms of cut thickness. ................................................... 58 Table 3.8 Regression and correlation analysis of the measured thrust force
component Ftm in terms of cut thickness..................................................... 59 Table 3.9 Regression and correlation analysis of the measured power force
component Fcm with rake angle................................................................... 60 Table 3.10 Regression and correlation analysis of the measured thrust force
component Ftm with rake angle. ............................................................... 61 Table 3.11 Linear regression with covariance analysis of the measured forces
components vs cutting speed V under different cut thickness t................ 62 Table 3.12 Covariance analysis of the effect of wearland size on the slopes of
measured power force Fcm vs cut thickness t ........................................... 63 Table 3.13 Covariance analysis of the effect of wearland size on the slopes of
measured thrust force Ftm vs cut thickness t............................................. 64 Table 3.14 Correlation analysis of chip length ratio on cut thickness. ...................... 66 Table 3.15 ANOVA for the effect of rake angle on chip length ratio. ...................... 67 Table 3.16 ANOVA results for the relationship between cutting speed and shear
angle. ........................................................................................................ 69 Table 3.17 ANOVA results for the effects of tool flank wear on shear angle. .......... 70 Table 3.18 ANOVA for the effect of tool flank wear on the relationship between
shear stress and rake angle. ...................................................................... 71 Table 3.19 Correlation analysis of shear stress on cut thickness. .............................. 72 Table 3.20 ANOVA for the effect of cutting speed on shear stress........................... 73 Table 3.21 ANVOA for the effect of tool flank wear on shear stress........................ 74 Table 3.22 Correlation analysis of friction angle against cut thickness t................... 75 Table 3.23 ANOVA for the effect of rake angle γ on friction angle β ...................... 76 Table 3.24 Results of ANOVA for the effect of cutting speed on friction angle β. .. 77 Table 3.25 Results of ANVOA for the effect of tool flank wear on friction angle β.78 Table 3.26 Coefficients in shear angle relation under different wearland sizes. ....... 79 Table 3.27 Correlation analysis of wearland force with process variables................ 82 Table 3.28 Regression results for rubbing forces with VB, V and rake angle by SPSS
.................................................................................................................. 83 Table 3.29 Correlations of the basic cutting quantities with cutting variables. ......... 86 Table 3.30 Regression Model Summary and ANOVA.............................................. 87 Table 3.31 Database for basic cutting quantities. ...................................................... 87 Table 3.32 Coefficients of the regression lines.......................................................... 88 Table 4.1 Parameters used in oblique cutting tests .................................................. 103 Table 5.1 Cutting parameters for turning operation tests......................................... 144
XI
STATEMENT OF ORIGINAL AUTHORSHIP This work contained in this thesis has not been previously submitted for a degree or
diploma at any other higher education institution. To the best of my knowledge and
belief, the thesis contains no material previously published or written by another
person except where due reference is made.
Signed:__________________________
Date:____________________________
XII
ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr Jun Wang for his meticulous
supervision, continuous encouragement, expert advice and financial assistance
throughout the entire course of this research. He also wishes to thank Associate
Professor Andy Tan for his valuable suggestions and guidance, particularly his
supervision in the final stage of this project.
Thanks are also due to Associate Professor Vladis Kosse (my associate supervisor)
for his assistance, to Mr David Golden, Mr Terry Beach, Mr Wayne Moore and Mr
Jonathan James for their enthusiastic assistance in the experimental work, and to
Professor Chuanzhen Huang for his proof-reading of some of the thesis chapters.
In addition, the author wishes to thank the staff in the School of Mechanical,
Manufacturing and Medical Engineering (currently the School of Engineering
Systems) and the Research Students Centre at the Queensland University of
Technology for providing an encouraging and stimulating research environment for
carrying out this project. The resources provided by the School of Mechanical and
Manufacturing Engineering at The University of New South Wales in the final stage
of this research are greatly appreciated.
Finally, and more importantly, the author wishes to thank Dr Hua Liu, for her
enthusiastic assistance.
XIII
NOMENCLATURE
Symbol Quantity
Aj area of chip section j for j=A, B (mm2)
b width of cut (mm)
b’ effective contact length of cutting edge in oblique cutting
(mm)
b* equivalent width of cut (mm)
bstraight length of the straight part of the side cutting edge engaged in
cutting (mm)
C1 and C2 material dependent constants
c1 and c2 constants
ce edge force intensity (N/mm)
Cce and Cte edge force intensity factors (N/mm)
Ccw and Ctw wearland force intensity factors (N/mm2)
Cs approach angle (Degree)
C’s projection of approach angle Cs on the rake face (Degree)
C*s equivalent approach angle (Degree)
CsL local approach angle on the chip element db edge element
length (Degree)
Csr local approach angle on any part of the nose radius edge
(Degree)
d’ projection of depth of cut d on the rake face (mm)
dA the area of the small chip element (mm2)
dF resultant force of dFx and dFy (N)
|dF0| magnitude of the friction force (N)
dFx,dFy,dFz elemental total forces at feed, radial and cutting speed
directions respectively (N)
dFcw,dFtw,dFrw element wearland forces at feed, radial and cutting speed
directions respectively (N)
XIV
dFox,dFoy friction force components on the rake in the X and Y
directions (N)
dFxe,dFye,dFze element edge forces at feed, radial and cutting speed directions
respectively (N)
dFxs, dFys,dFzs element forces required for the basic chip formation at feed,
radial and cutting speed directions respectively (N)
dFxw,dFyw,dFzw element wearland forces at feed, radial and cutting speed
directions respectively (N)
f feed rate in turning operation (mm/r)
Fo resultant friction force (N)
Fcm measured power forces at shear zone (N)
Ftm measured thrust forces at shear zone (N)
Fcs power forces at shear zone (N)
Fts thrust forces at shear zone (N)
Fce edge force along power forces direction (N)
Fte edge force along thrust force direction (N)
Fe edge force (N)
Fcint rubbing force component along the power direction (N)
Ftint rubbing force component along the thrust force direction (N)
Fcrub rubbing force component along the power direction (N)
Ftrub rubbing force component along the thrust force direction (N)
Fw wearland force (N)
Fcw weraland force component along the power force direction (N)
Ftw wealrand force component along the thrust force direction (N)
Fs total forces for chip formation (N)
Frs radial force in oblique cutting (N)
Fce’ edge force component normal to the tool edgy in oblique
cutting (N)
Fte’ edge force component normal to the cutting speed direction in
oblique cutting (N)
Fre edge force in radial direction (N)
Frw wearland force at radial direction (N)
Fc total predicted power force in oblique cutting (N)
Ft total predicted thrust force in oblique cutting (N)
XV
Fr total predicted radial force in oblique cutting (N)
Fx, Fy ,Fz total forces at feed, radial and cutting speed directions
respectively (N)
Fxs, Fys, Fzs forces required for the basic chip formation at feed, radial and
cutting speed directions respectively (N)
Fxe, Fye, Fze edge forces at feed, radial and cutting speed directions
respectively (N)
Fxw,Fyw, Fzw wearland forces at feed, radial and cutting speed directions
respectively (N)
Fxe1,Fye1, Fze1 edge forces on the straight part of the cutting edge in X, Y and
Z directions (N)
Fxe2,Fye2, Fze2 edge forces on the nose radius edge in X, Y and Z directions
(N)
Fxw1,Fyw1, Fzw1 wearland forces on the straight part of the cutting edge in X, Y
and Z directions (N)
Fxw2,Fyw2, Fzw2 wearland forces on the nose radius edge in X, Y and Z
directions (N)
h tool-chip contact length. (mm)
h’ length on a constant shear stress acts (mm)
i inclination angle (Degree)
i* inclination angle of equivalent cutting edge (Degree)
iL local inclination angle on the chip element (Degree)
ir local edge inclination angle on any part of the nose radius edge
(Degree)
K the shear stress ratio on the shear plane
Kce, Kte and Kre edge force coefficients per unit width of cut in the oblique
cutting (N)
Kcs chip forming power force coefficients in oblique cutting
(N/mm2)
Kts chip forming thrust force coefficients in oblique cutting
(N/mm2)
Krs chip forming radial force coefficients in oblique cutting
(N/mm2)
Kc slope (N/mm)
XVI
Kt slope (N/mm) *csK , *
tsK , *rsK coefficients (N/mm2)
lc chip length (mm)
L the tool-chip contact length ratio to cut thickness
P*s tool cutting edge plane for equivalent cutting edge
P*n cutting edge normal plane for equivalent cutting edge
Pr reference plane
Pp tool back plane
Pf work plane
r tool nose radius (mm)
rl chip length ratio
R total resultant force (N)
Rs resultant shear force (N)
Re resultant edge force (N)
Rw resultant wearland force of Fcw and Ftw (N)
Re’ resultant edge force in oblique cutting (N)
t cut thickness (mm)
t* equivalent cut thickness (mm)
u friction force intensity on the rake face
VB wearland size (mm)
V cutting speed (m/min)
VB’ effective wearland size in oblique cutting (mm)
Wc weight of chip (mm)
XVII
Greek symbols
Symbol Quantity
β friction angle (Degree)
βn normal friction angle (Degree)
φ shear angle (Degree)
φn normal shear angle (Degree)
γ tool (normal) rake angle (Degree)
γn normal rake angle (Degree)
γ*n equivalent normal rake angle (Degree)
c chip flow angle (Degree)
η0 angle between the equivalent cutting edge and straight part of
the side cutting edge in the rake face plane (Degree)
η’0 projection of η0 in the reference plane Pr (Degree)
angle made by positive x axis with the undeformed chip
element (Degree)
1, 2, 3 limits of integration (Degree)
ϕ acute angle between the shear plane and resultant cutting force
(Degree)
ρ Density of work material (g/mm3)
τ shear stress (N/mm2)
0 angle made by dFo with the positive Y axis (Degree)
0Ω chip flow angle with reference to the major cutting edge
(Degree)
µ coefficient of friction in the rake face
1
Chapter 1 Introduction
1.1 Background The conventional material removal process has long been accepted as a major
manufacturing process due to its inherent advantage and technological capabilities.
One major objective is to continually improve the technological and the economic
performance of the process as assessed by the force, power, component surface finish
and dimensional accuracy, tool wear and tool-life as well as production rate and cost
[1].
The importance of machining performance information such as the forces, power and
tool life has long been recognized. Such information is essential for the design and
selection of machine tools and cutting tools as well as the optimisation of cutting
conditions for the efficient and effective use of the machining operations. The
globalisation of the manufacturing industry and activities has resulted in greater
demands on high product quality and low production costs with unmanned machining
operations. The untended manufacturing operations including NC/CNC machining
need to operate at optimum efficiency necessitating on-line tool condition monitoring
and improved adaptive control. Thus, further economic gains in modern
manufacturing systems will have to rely on the improvement of machining
operations, as the available production time is being used to almost full capacity
considering the time required for maintenance and other non-production activities.
Therefore, a reduction in production cost and an increase in productivity can be
realized by optimising the cutting conditions and making the most use of a tool’s life.
This has generated greater demands for reliable machining performance information
and realistic machining optimisation strategies for further economic gains.
Turning operation is a very important material removal process in modern industry.
At least one fifth of all applications in metal cutting are turning operations [2].
Turning operation is also an essential single-point-tool material removing process for
further studying milling and drilling as well as other cutting operations [3]. Although
the empirical approach has been used to experimentally relate the various process
2
variables to the different machining performance measures or indicators, the
fundamental cutting approach can provide alternative and complementary means of
arriving at quantitative predictions of machining performance. Since the
establishment of the fundamental or mechanics of cutting approach in the middle of
the last century, orthogonal cutting has been recognized as the basic cutting process to
study the oblique cutting and turning operation using a single point tool [1].
In a machining process, the cutting tool is not perfectly sharp; there is always an edge
radius at the cutting edge of a ‘sharp’ cutting tool. In addition, tool wear is a natural
and inevitable process for all cutting tools although the wear rate may be controlled or
optimized by varying the cutting conditions according to the process requirements.
Since the establishment of the orthogonal cutting model and associated theories by
Merchant in early 1940s, several versions of orthogonal cutting models have been
developed. It has been found that the thin shear zone model significantly simplifies
the analysis and can give adequate descriptions of the orthogonal cutting process [4].
However, it was soon found that all the orthogonal cutting models including the thin
shear zone model have been found to have limitations in predicting cutting
performance. It was found that a rubbing or ploughing process could occur in the
vicinity of the cutting edge resulting in an edge force in addition to the force due to
the chip formation given in the thin shear zone analysis [4-6]. This force is thought to
be manifested by the positive force intercepts when the measured force versus cut
thickness graphs are extrapolated to zero cut thickness, and is proportional to the
engaged cutting edge length [5]. The edge force can greatly affect the basic cutting
quantities, such as the shear stress and friction angle, obtained from the cutting model
and the measured forces. Some investigations and experimental data have supported
the existence of the edge force and it has been suggested that the edge force to be
from the measured force data when evaluating the basic cutting quantities using the
shin shear zone model [5].
It should be noted that tool wear normally occurs on the flank (flank wear) and rake
face (crater wear). It has been reported that the large dragging or rubbing force on the
flank wearland is responsible for total force increase, while the crater wear on the
rake face in fact results in a slight decrease in the cutting forces [4]. It is believed that
3
a tool with crater wear behaves like a restricted contact tool and hence the force
decreases. Crater wear occurs at relatively higher cutting speeds and becomes
negligible compared with the flank wear at lower speeds [7]. Under many cutting
conditions, when modern coated cutting tools are used, flank wear becomes much
more important in practice than the crater wear. The crater wear mainly influences the
chip-flow on the tool rake face [8-10]. For these reasons, this project will be limited
to considering the tool flank wear only.
1.2 Objectives of the Research From the above background analysis, the objectives of the research are as given
below:
(a) A comprehensive literature survey is required to understand the current
development in this field and to justify the objectives and define the scope of
this project.
(b) The effects of tool flank wear on the orthogonal cutting process then needs to be
investigated by an experimental investigation involving orthogonal cutting tests
and statistical analysis of the experimental data. Special attention should be paid
to assessing if the flank wear affects the basic cutting or chip formation process
in the shear zone. This will be done by examining the effect of flank wear on the
basic cutting quantities (e.g. shear angle, friction angle and shear stress). If the
flank wear does not affect the chip formation process in the shear zone, the well
developed orthogonal cutting models and the associated theories, such as the
thin shear zone analysis, can be readily applied to the study of cutting processes
when tool flank wear is present. The thin shear zone analysis in orthogonal
cutting for sharp tools needs to be reconsidered with a view to incorporating the
tool flank wear effect in modelling the cutting performance [1, 11, 12]. For the
purpose of this study and due to the time limitations on this project, the rubbing
or ploughing force on the wearland (or wearland force) was developed based on
the experimental data.
(c) The orthogonal cutting force models allowing for the tool flank wear need to be
developed first to assess the new modelling approach, which uses the existing
4
cutting theories with the addition of wearland forces in predicting the cutting
forces.
(d) The orthogonal cutting force models will need to be extended to a more general
cutting process, i.e. the classical oblique cutting, to assess the validity of the
new modelling approach.
(e) Finally, the predictive cutting force models for practical turning operations
when tool flank wear is present will be developed for general lathe tool
geometry with a nose radius and under oblique cutting conditions. The models
will be qualitatively and quantitatively assessed and verified.
1.3 Thesis Organisation The thesis is organised into 6 chapters. In Chapter 2, the motivation and the aim of
the current project is highlighted with reference to the comprehensive literature
review. In Chapter 3, an experimental investigation of the orthogonal cutting process
with tool flank wear is presented first. Based on the findings of this investigation, a
cutting force model for orthogonal cutting with tool flank wear is then proposed and
verified. A database for the basic cutting quantities is developed and presented
together with the edge forces and wearland forces which are required for developing
the cutting force models in the ‘classical’ cutting as well as practical turning
operations. In Chapter 4, the oblique cutting force model is proposed which allows
for the effects of tool flank wear. The model is then verified by an experimental
study and by statistically comparing the model predicted with experimental cutting
forces obtained. In Chapter 5, the predictive cutting force models for turning
operations are developed and verified. The models consider the general tool
geometry with a round corner edge under both orthogonal and oblique cutting
conditions, and allow for tool flank wear effects. Finally, Chapter 6 presents the main
project findings as well as the achievements and contributions of the project. Future
work is also recommended in this chapter.
5
Chapter 2 Literature Review
Both the empirical approach and mechanics of cutting analysis are used in the study
of cutting performances. In the empirical approach, experimentally measured
machining characteristic values such as the forces and tool-life are related to cutting
conditions by regression analysis. This approach involves considerable testing to
determine the constants in the empirical equations and the results apply only to the
machining operation tested. There appears to be a distinct dearth of fundamental
studies of cutting characteristics for machining with tool wear and the associated
predictive models for technological performance measures. Nevertheless, a number of
important studies have been reported on developing the relationship between tool
wear and cutting forces and power in milling operations using the ‘mechanistic’
approach favoured in North America [13-18]. In this approach, special machining
tests are conducted to establish ‘force coefficients’ relating the measured forces to the
cut proportions by empirical equations for the tool geometry. These are then used to
develop models for predicting cutting forces and power for the ‘calibrated’ tool over a
range of cutting conditions. A recognized drawback of this approach, as noted in the
milling work [13, 19, 20], is that only two of the three force components have been
included in the models, assuming these are related by a ‘constant’ ratio for a given
tool geometry, and that each tool geometry has to be individually tested to establish
the ‘force coefficients’ needed in the force model.
Empirical approach method is extremely time consuming and costly. This becomes
especially important when the constant introduction of new work and tool materials is
considered and when it is realized how even small deviations from the normal
composition of a work material could cause large changes in its machining
characteristics. Given the significantly large and unmanageable number of tool-work
material combinations, cutting and tool variables and different practical machining
operations, an empirical approach is clearly undesirable in practice. A fundamental
approach and associated cutting models rely less on the experimental testing and are
more suitable in practice.
Although a considerable amount of work has been done on the fundamental studies of
machining performances in turning and other machining operations based on various
6
approaches (e.g. [5, 19-30]), research on cutting forces incorporating tool wear effects
has received little attention [13-15, 31]. An underlying objective of most of these
reported investigations was to establish the relationship between the tool wear and
cutting forces. In these studies only the function of the cutting force and tool wear
relationship has been established using experimental studies with specific cutting
condition to forecast tool wear or monitoring cutting tool condition [18, 32-38].
There are a limited number of studies on predictive models for turning operations
allowing for tool wear which have been published based on the mechanics of cutting
analysis. The approach used in the study is based on the mechanics of cutting analysis
for sharp tools, incorporating the ‘edge force’, and mathematically relates each
practical operation such as turning to the fundamental oblique cutting process [5, 24-
26, 39]. With this approach, only the basic cutting data (shear angle, friction angle
and shear stress) from the orthogonal cutting tests are required for the relevant tool-
work material combinations.
For machining with tool wear, the basic assumption is that tool flank wear does not
statistically affect the basic cutting quantities (shear angle, friction angle, shear stress
etc.), as compared to sharp tool cutting, but produces additional rubbing forces on the
wearland on the tool flank surface and the workpiece. This assumption will be
confirmed by the orthogonal cutting tests and the data analysis. Consequently, the
mechanics of cutting analysis for sharp tools can be further developed to allow for the
additional rubbing or wearland force in total force prediction, once the basic wearland
force quantities are achieved from the orthogonal cutting tests. The cutting force
models allowing for the tool flank wear will be further extended to an oblique cutting
and turning operation with worn tool cutting conditions.
An extensive literature survey on the mechanics of cutting analysis and predictive
force models for orthogonal and oblique cutting as well as the more general practical
machining operations has been carried out in this chapter.
7
2.1 Orthogonal and oblique cutting models for perfect sharp tools
Since the establishment of the orthogonal shear plane model of Ernst and Merchant
[40, 41] in the 1940’s and the slip-line field model of Lee and Shaffer [42] in the
1950’s, later Oxley [43] and Armargeo [44], a large amount of work has been carried
out on the fundamental studies of machining performances in orthogonal, oblique and
turning operation. Numerous investigations have confirmed that an understanding of
the classical orthogonal cutting is essential for modelling the various practical
machining operations such as turning and milling.
The shear zone in the orthogonal cutting process has been recognised as ‘thin’ zone
by some researches and ‘thick zone’ by others. Various analyses have been found in
the literatures based on both shear zone models. The thick zone model was studied by
Palmer and Oxley [45], von Turkovich [46] and Okushima and Hitomi [47]. The
experimental investigation conducted by Oxley has shown that even though the shear
zone appeared thick, the measured deformation, shear strains and strain rates in the
shear zone were located in a narrow region about the ‘idealised shear plane’ [48]. It
can be concluded that the ‘thick shear zone’ could be adequately represented by a thin
shear zone. On the other hand, the thin zone model was suggested by Merchant [12],
Lee and Shaffer [42], Armarego and Brown [4]. It has been identified as more
representative of the physical cutting phenomena than the thick zone model [5]. Also
the thin zone analysis has served a useful purpose by qualitatively explaining the
experimentally observed behaviour in cutting. Based on the shear plane model of
Ernst and Merchant [40, 41], comprehensive mathematical force predictive models
for various machining operations [4, 5, 43, 44] have been developed for ‘sharp’ tools.
2.1.1 Orthogonal cutting models for sharp tools
In the fundamental machining investigations, primary attention has been paid to
modelling the geometrically simple orthogonal cutting process involving two-
dimensional plastic deformation [3, 4, 11, 12, 49, 50], as shown in Fig. 2.1.
8
Fts
Fcs
N
F
R
F
N
R’
β
β
Vs
Fn Fs
φ
γ
t
Workpiece
Tooltc
Chip
Figure 2.1 Model of chip formation in orthogonal cutting for a ‘sharp’ tool.
The geometrical relationship of orthogonal cutting is shown in Fig. 2.1. Cutting force
relationships are given by [44]: (The symbols are defined in the Nomenclature.)
Fbt
cs =−
+ −τ β γφ φ β γ
cos( )sin cos( )
(2.1)
Fbt
ts =−
+ −τ β γφ φ β γ
sin( )sin cos( )
(2.2)
The force components have been shown to be dependent on the width of cut b and
cut thickness t, which control the interference area of cut. They are also dependent on
the tool rake angle γ, the deformation geometry during chip formation depicted by
the shear angle φ, and the work material property τ, as well as friction properties on
the rake face depicted by the friction angle β. In the investigation of orthogonal
cutting, the measured forces Fcs and Fts increase linearly with cut thickness t and
cutting width b, while chip length ratio rl, shear angle φ and shear strain τ were found
to be independent of b and t in orthogonal cutting.
9
To predict the cutting forces Fcs and Fts from the above equations, for a given cutting
variables: b, t, γ, and V for any cut, it is necessary that the shear stress τ, the friction
angle β and the shear angle φ must be known. Since the shear stress and friction
angle could be considered to be basic quantities, they could be estimated from the
published conventional material test and friction data. The shear angle represents the
deformed geometry during chip formation and must be found.
Considerable research has been carried out in attempting to establish a mathematical
relationship between the shear angle at shear zone and the friction angle at the rake
face in the classical orthogonal cutting. The shear angle relationship formed a part of
the mechanics of cutting analysis, which enables the shear angle to be prediction by a
given or known friction angle and tool rake angle. Various theoretical relationships
between the shear angle and the friction angle and the rake angle have been
proposed.
It is assumed the shear angle adjusted itself during any one cut, when minimum
energy principle applied to the cutting process and by differentiating equation (2.1)
with respect to the shear angle, obtained the following relationship between shear
angle, the rake angle and the friction angle [44]:
( )γβφ π −−= 21
4 (2.3)
The equation 2.3 represents an important step in the mechanics of cutting analysis.
But experimental tests showed the shear angle relationship in the above equation
agree unsatisfactorily with experimental results. The coefficient of friction value was
also found to be higher than sliding friction coefficients given in handbooks.
Merchant [41] modified the shear angle relationship assuming the shear flow stress
on the shear plane to be dependent on the normal stress for shear plane in order to
account for his experimental findings. Accordingly, the shear angle relationship is
shown as:
( )βγφ −+= − k121 cot (2.4)
Where k is the slope of the plot between shear stress versus the normal stress on the
shear plane, and is considered a ‘machining constant’. The introduction of
‘machining constant’ showed an improvement in correlation with experimental
values. The friction angle, shear stress and ‘machining constant’ must be found from
10
orthogonal cutting tests before the forces and power could be predicted. But there are
some doubts on the quantitative validity of Merchant’s new shear angle relationship.
Lee and Shaffer [42] applied the theory of plasticity for an ideal rigid-plastic material
for orthogonal cutting process assuming the same thin shear plane deformation. They
considered the plastic flow to occur in the slip lines and the rigid chip to occur in a
triangular plastic field. The shear angle relationship can be shown.
( )γβφ π −−= 4 (2.5)
The correlations between Lee and Shaffer’s shear angle relationship and
experimental results have been found to be minimal.
Shaw, Cook and Finnie [51] modified the Lee and Shaffer’s model by assuming that
the shear plane was not a plane of maximum shear stress. The shear angle
relationship is expressed as:
( )γβηφ π −−+= )'( 4 (2.6)
A variation to the shear angle relationship was made by the introduction of η‘, which
takes into account the variation of different workpieses. This model is adequate,
because it introduces another unknown variable η’. Further, Shaw et al. could not
provide a theoretical analysis to justify or estimate the value η’. The basic weakness
of the model lies in the assumption that the shear plane is not the maximum shear
stress plane.
Oxley [43, 52] proposed an alternative model for the shear angle relationship. The
analysis showed that the hydrostatic stress at the shear zone was linearly distributed.
While shear stress on the straight slip line was uniformly distributed. The analysis is
based on a parallel sided thin shear zone model of finite thickness, a slip line theory,
which allowed for variations in the shear stress due to work-hardening and uniform
stress distributions at the rake face. From the shear distributions at the shear zone and
at the tool-chip interface including chip equilibrium, the equation is given by [52]:
11
2
)(2sintan2
)(2cos42
1(tan 1 γφ
βγφφπϕ −−−+−+= − (2.7)
Where: ϕ is the acute angle between the shear plane and resultant cutting force.
Similarly from chip equilibrium based on Merchant’s model, the resultant cutting
force R on the chip applied at the shear plane was considered to be equal, opposite in
direction and collinear to the force R’ applied to the chip at the tool-chip interface.
The second relationship is derived and given by:
( )γβφϕ −+= (2.8)
By equating the above two equations, ϕ can be eliminated and results in an intricate
relationship between the shear angle, friction angle and rake angle. The shear angle
can only be solved by numerical methods for a given rake angle and friction angle.
The solution is rather complex. Therefore, non-uniform normal and tangential stress
distributions on the rake face were assumed in the second analysis. These rake face
distributions resulted in a modified form of equation (2.7) and given by [43]:
)2
)(2sintan2
)(2cos)1(
421
(tan'
1 γφβ
γφφπϕ −−−++−+= −
hh
(2.9)
Where: h is the tool-chip contact length.
h’ is the length on which a constant shear stress acts i.e. ‘sticking’ friction.
In this model, h’/h could be estimated by numerically solving equations (2.8) and
(2.9). Then the shear angle can be found. When the experimental value of (β-γ) from
previous studies [41] were plotted against φ evaluated using equations (2.8) and
(2.9), it was found that the shear angle relationship showed a linear relationship for
the various values of h’ and h. Although this shear angle relationship was claimed
[43] to agree reasonably well with Merchant’s data [12] for steel work materials, it is
difficult to evaluate h’ and h in the equations. It also requires the friction angle to be
found from experimental tests before the shear angle can be predicted. Furthermore,
the shear angle relationship for other work materials was not tested in his research
work. Armarego has found the correlation between the above shear angle
12
relationships and those found from orthogonal cutting test data for other work
materials to be only minimally related [53]. Thus model needs to be improved.
Other researchers have also attempted to establish the shear angle relationship and
are summarised as follows:
Huchs [54] in 1951 presented the relationship;
2
2tan4
1 µγπφ−
−+= (2.10)
Where: µ is the coefficient of friction in the rake.
Weisz [55] in 1957 presented the relationship as;
)(7.54 0 γβφ −−= (2.11)
Sata and Mizuno [56] approximated the shear angle as
γφ = or 15o (2.12)
Sata and Yoshikawa [57] expressed the shear angle relationship in 1963 as;
KL)sin(4
coscotcos
γθθθφ+
+= (2.13)
Where: K is the ratio of the shear stress on the shear plane to that on the rake face;
and L is the ratio of the tool-chip contact length to cut thickness.
All the relationships mentioned above require one or more parameters to be known in
order to predict the shear angle for a given tool geometry and cutting condition in
orthogonal cutting processes. Unfortunately, the various parameters included in the
shear angle relationships are not readily available or can only be theoretically
predicted.
After reviewing different forms of the shear angle relationships and experimental
results, Armarego and Brown [4] concluded that shear angle relationship could be
expressed in a more general form:
( )γβφ −−= 21 CC (2.14)
13
Where: C1 and C2 are material dependent constants.
The shear angel can be predicted by the equation 2.17. But experiments confirmed
that τ and β cannot be found from available handbook data and must be found from
orthogonal cutting tests. In addition, the parameters C1 and C2 have found to be work
material dependent and difficult to predict theoretically. The parameters C1 and C2
could be solved using experimental data and the shear angle relationship can be
found.
In practice, the shear angle, friction angle and shear stress can be found from a series
of orthogonal cutting tests for different work materials and cutting conditions by
measuring the cutting forces and chip length rates rl. The shear angle, friction angle
and shear stress are calculated by the following equations [44]:
−= −
γγφ
sin1cos
tan 1
l
l
rr
(2.15)
+== −−
cs
ts
FF11 tantan γµβ (2.16)
btFF tscs φφφτ sin)sincos( −= (2.17)
When the shear angle, friction angle and shear stress (basic cutting quantity database)
are found from the cutting tests, the power and thrust forces can be predicted for a
given cutting condition in the orthogonal cutting.
2.1.2 Oblique cutting models for sharp tools Although the idealized orthogonal process is a close approximation to many actual
machining operations, it is clear that chip-formation process occurring at the cutting
edges of most tools could be modelled more accurately by an oblique model in which
the cutting edge is inclined to the cutting velocity. The three-dimensional plastic flow
involving chip formation in oblique machining is far more complex than the relatively
simple two-dimensional orthogonal cutting process.
The thin shear zone model for ‘classical’ oblique cutting is an extension of the
orthogonal case using similar assumptions. The additional cutting variable and
inclination angle needs to be introduced. The inclination angle plays a major role in
14
controlling the chip flow direction and the resultant cutting force. From the force and
deformation considerations and allowing for the ‘collinearity conditions’ the
following equations have been developed [4]:
ncnnn
ncnn
ncs
itbF
βηγβφβηγβ
φτ
222 sintan)(cos
sintantan)cos(sin ⋅+−+
⋅⋅+−⋅⋅⋅= (2.18)
ncnnn
nn
nts i
tbF
βηγβφγβ
φτ
222 sintan)(cos
)sin(cossin ⋅+−+
−⋅⋅⋅= (2.19)
ncnnn
ncnn
nrs
itbF
βηγβφβηγβ
φτ
222 sintan)(cos
sintantan)cos(sin ⋅+−+
⋅−⋅−⋅⋅⋅= (2.20)
The shear angle and friction angle are given by:
ncl
ncln ir
irγη
γηφsin)cos/(cos1
cos)cos/(costan
⋅⋅−⋅⋅= (2.21)
cn ηββ costantan ⋅= (2.22)
The collinearity condition provides a means of relating the forces and velocities in
oblique cutting. The condition assumes the friction force on the rake face is collinear
to the chip velocity direction and shear force in the shear plane is collinear to the
shear velocity direction. When the collinearity conditions are satisfied the following
equation must be met to find the chip flow angle ηc [4, 5]:
i
i
nc
nnn tansintan
costan)tan(
⋅−⋅=+
γηγβφ (2.23)
From the above equations, the cutting forces can be predicted by the basic cutting
quantities φn (degree), βn (degree), τ (N/mm2) and ηc (degree), for the given cutting
conditions, γn (degree), b (mm), t (mm) and i (degree).
Another oblique cutting analysis has been proposed by Lin [58]. The oblique cutting
analysis is based on the assumption that the oblique cutting process can be
represented as a combination of two processes, namely, an orthogonal cutting
process in the normal plane and another friction process in the tool-chip interface
including a shearing process in the shear plane. The directions are normal and
15
parallel to the cutting edge. It was also assumed that the friction force in the rake
face in oblique cutting was collinear to the chip velocity. But the shear flow along
the cutting edge direction in the shear plane was ignored by assuming the shear flow
angle to be zero in analysis. In fact, the shear flow angle in the shear plane is
generally not zero in the oblique cutting. In the analysis, the friction angle β0 was
found from orthogonal cutting tests and the shear stress was represented by empirical
equations [58]. This can be considered as an ‘equivalent orthogonal cutting’ analysis.
According to Lin’s model, the forces in the oblique cutting can be predicted for a
given φ0, β0, τ and ηc, when the cutting condition γn, b, t and i are known. The cutting
force Fts should be increased with the increase in the inclination angle. However, this
is contradictory to the trend of the thrust force which is not changed with the angle of
inclination (up to about 40 degrees) found from literature [4, 5]. There is not enough
evidence to support the equivalent orthogonal cutting in the normal plane of oblique
cutting. Thus simplified oblique cutting analysis is considered unsuitable for this
project.
2.2 Orthogonal and oblique cutting models incorporating edge force
Generally, a cutting tool is not perfectly ‘sharp’. There is always an edge radius on
the cutting edge. Hence, there are ‘edge forces’ on the cutting edge [44]. It has been
suggested [4-6] that since the cutting edge is not perfectly sharp, a rubbing or
ploughing process could occur in the vicinity of the cutting edge resulting in an edge
force in addition to the force due to the chip formation given in the thin shear zone
analysis. This force is thought to be manifested by the positive force intercepts when
the measured force versus cut thickness graphs are extrapolated to zero cut thickness,
and is proportional to the engaged cutting edge length [5, 51, 53, 59, 60]. The edge
force can greatly affect the basic cutting quantities, such as the shear stress and
friction angle, obtained from the cutting model and the measured forces. Some
investigations and experimental data have supported the existence of the edge force
and it has been suggested to remove the edge force from the measured force data
when evaluating the basic cutting quantities using the shin shear zone model [5].
16
2.2.1 Orthogonal cutting models incorporating edge force
Extensive studies have been carried out at the University of Melbourne on the
experimental cutting characteristics with the change of input cutting variables, which
include the cut thickness, width of cut, rake angle and cutting speed for a number of
materials. The results show there is strong evidence of the existence of a
concentrated ‘edge force’ [5, 53, 59]. The shear angle has been found to be
independent of the width of cut b and cut thickness t. Also results indicated that the
shear angle was unaffected by cut proportion and size, and that the friction angle
should be independent of b and t because of the existence of a shear angle
relationship. The force components should be dependent on b and t, and increase
linearly with b and t. But the orthogonal cutting tests showed that there were
significant positive intercepts when t is reduced to zero. The typical trends showed
the constant deformation geometry (shear angle), linear slopes of the force
components and consistent values of friction angle and shear stress. They were
analysed after the intercepts were removed. The intercept forces were the rubbing
forces on the tool edge, which were independent of chip formation. Alternative
explanations that the ‘intercept forces’ were part of the chip formation process have
been found to be deficient [44, 48]. Thus the measured forces can be expressed by a
linear equation with positive intercepts, which are the combinations of the forces (Fcs
and Fts) at shear zone and edge forces (Fce and Fte) at the cutting edge. The measured
force equations are given by [5]:
cecscm FFF += (2.24)
tetstm FFF += (2.25)
The edge forces are expressed by:
bCF cece = (2.26)
bCF tete = (2.27)
Where: Cce and Cte (N/mm) are the edge force coefficients.
Consequently, when the edge forces have been removed from the measured forces,
the basic cutting quantities, such as friction angle and shear stress, obtained from the
cutting test can represent the cutting process. The friction angle β and shear stress τ,
Equations (2.4) and (2.5) have been modified and shown below:
17
−−+= −
cecm
tetm
FFFF1tanγβ (2.28)
( ) ( )[ ]
btFFFF tetmcecm φφφτ sinsincos −−−= (2.29)
The ‘modified thin zone’ model incorporating the ‘edge force’ has been proposed
after the experimental trends and basic cutting parameters have been analysed and
evaluated [5].
bCbt
FFF cececsc +−+
−=+=)cos(sin
)cos(γβφφ
γβτ (2.30)
bCbt
FFF tetetst +−+
−=+=)cos(sin
)sin(γβφφ
γβτ (2.31)
This model provides practical functions to predict the cutting forces in orthogonal
cutting incorporating the effect of cutting edge. According to the model, the cutting
forces can be predicted by the basic cutting quantities (φ, β, τ) and the edge force
coefficients (Cce and Cte). The database of the cutting quantities (φ, β, τ, Cce and Cte)
can be found from a number of orthogonal cutting tests over a range of cutting
conditions for given tool and workpiece combinations.
2.2.2 Oblique cutting models incorporating edge force
In general, in oblique cutting, the characteristics have shown similar trends to those
of classical orthogonal cutting where comparisons were possible. Thus the effects of
t, b and γn can be readily compared for the two operations while the effects of angle
of inclination are in fact not comparable except that orthogonal cutting represents the
special case of oblique cutting with i=0. The experimental trends in the oblique
cutting tests [44] show that the chip length ratio rl and normal shear angle φn are
independent of width of cut b and cut thickness t for given normal rake angle γn and
cutting speed. The measured force components Fcm, Ftm and Frm in oblique cutting
are directly proportional to the width of cut b and cut thickness t. The effect of the
cut thickness t on the measured force showed a linear relationship with a positive
intercept. The basic cutting quantities were calculated by treating the intercepts as
‘edge’ forces and subtracted from the measured forces, which were found to be
18
independent of cut thickness t. The mechanics of cutting analysis for oblique cutting
[4] has been modified to incorporate the significant intercept forces as ‘edge’ forces
[5, 44]. Accordingly, the three forces in oblique cutting have been represented as the
sum of the forces for the chip formation and the ‘edge’ forces, which are proportional
to the width of cut. The equations in oblique cutting were given by [44]:
bKFFFF cecscecscm +=+= (2.32)
bKFFFF tetstetstm +=+= (2.33)
bKFFFF rersrersrm +=+= (2.34)
Where: Kce, Kte and Kre (N/mm) are the edge force coefficients per unit width of cut
in the oblique cutting.
Accordingly, the predicted cutting forces in oblique cutting are expressed by:
bKitb
F ce
ncnnn
ncnn
nc +
⋅+−+⋅⋅+−⋅⋅⋅=
βηγβφβηγβ
φτ
222 sintan)(cos
sintantan)cos(sin
(2.35)
bKi
tbF te
ncnnn
nn
nt +
⋅+−+−⋅⋅⋅=
βηγβφγβ
φτ
222 sintan)(cos
)sin(cossin
(2.36)
bKitb
F re
ncnnn
ncnn
nr +
⋅+−+⋅−⋅−⋅⋅⋅=
βηγβφβηγβ
φτ
222 sintan)(cos
sintantan)cos(sin
(2.37)
The cutting force can be predicted by the above equations, if the basic cutting
qualities and cutting conditions are known in the oblique cutting. Detailed
experiments and analysis for the oblique cutting indicated that the basic cutting
quantities, rl, β and τ were unaffected by the inclination angle in the range of 0o~40o,
at a constant normal rake angel γn, cutting speed V and tool and workpiece
combination conditions [5]. This indicated that the basic cutting quantities, rl, β, and
τ found in orthogonal cutting tests could represent the values in oblique cutting under
the same cutting conditions. The edge force coefficients Cce and Cte in orthogonal
cutting were directly used in the oblique cutting to calculate the edge forces (i.e.
Kce=Cce and Kte=Cte). The edge force component, Fre was neglected under conditions
where the inclination angle is less than 40 degree [5].
19
Other edge force coefficients proposed by Whitfield for oblique cutting are given by
[26]:
iCK cece cos= (2.38)
tete CK = (2.39)
iCK cere sin= (2.40)
In general, the above edge force models can be considered adequate as they can be
extended appropriately and represent a practical machining operation [5].
2.3 Predictive force models for turning operations
The turning operations is the most commonly used machining process. The basic
cutting action and phenomena are similar to the classical orthogonal or oblique
cutting processes at low feed to depth of cut ratio (f/d). In turning operations, the feed
is much smaller than the depth of cut (f<d) and the depth of cut is much smaller than
the workpiece diameter (d<<D). The feed speed Vf is also much smaller than the
tangential velocity V. The resultant cutting velocity can be considered as
approximately constant and perpendicular to the tool axis and base. The angle of
inclination i at the major cutting edge of a common lathe tool can be used to identify
the turning process as ‘equivalent orthogonal cutting’ (when i =0) or ‘equivalent
oblique cutting’ (when 0≠i ). In common turning operations, the feed f is akin to the
cut thickness t and the depth of cut is comparable to the width of cut b in the
‘classical’ single edge wedge tool processes. Hence, the cutting force models in
orthogonal and oblique cuttings can be used in turning operations.
2.3.1 Cutting force models for single straight cutting edge
There are three main cutting forces in a turning operation, namely the feed force Fx,
radial force Fy and the power force Fz. The feed force is parallel to the tool transition
and work-piece axis, which controls the load on the feed mechanism. The radial
force induces the deflections on the tool holder and workpiece in the relative
direction and controls the accuracy of the workpiece. The power force is parallel to
the cutting speed direction and controls the power and torque of the main spindle.
20
The fundamental or mechanics of cutting approach to cutting force has been
developed from the mathematical model of ‘classical’ orthogonal and oblique cutting
processes [4]. The lathe tool has been approximated to the tool in oblique cutting by
assuming the depth of cut (d) is larger than the feed rate (f) in many turning
operations. The cutting process occurs mainly at the straight major cutting edge.
When the tool tip radius is small compared to the depth of cut, the tool in the turning
operation may be approximated to the tool in oblique cutting. When the tool corner
radius was ignored, the forces in the turning operation can be determined by using of
the mechanics of cutting analysis in oblique cutting.
In turning operation, the depth of cut (d) and feed rate (f) can be related to cut
thickness (t) and width of cut (b) in oblique cutting:
sCdb cos/= (2.41)
sCft cos= (2.42)
btdfA =≈ (2.43)
Where: Cs (degree) is the approach angle and A is the cutting area.
The force components Fz , Fy and Fx found from the oblique cutting force
components can be expressed as:
cz FF = (2.44)
srsty CFCFF cossin −= (2.45)
srstx CFCFF sincos += (2.46)
Where: the Fc, Ft and Fr (N) are the oblique cutting force with the edge forces.
It is expected that the cutting forces can be calculated for a given cutting conditions
and tool specified angles, while the basic cutting quantities and edge forces
coefficients can be found from the database for the relevant work-material obtained
from ‘classical’ orthogonal cutting tests.
The predicted force trends in a turning operation have been studied and compared to
those in a classical oblique cutting [4, 44]. The forces increased with depth of cut and
the forces increase linearly with the feed rate (f) which exhibited positive force
21
intercepts at f=0. It represents edge forces due to ‘rubbing’ or ‘ploughing’ at the
cutting edge. This was also noted in the ‘classical’ oblique cutting forces with respect
to the width of cut b and cut thickness t, when t=0. Similarly, it was noted that the
normal rake angle γn controlled the power force component Fz in turning operations.
The power force is independent of the inclination angle (i) and the approach angle
(Cs).
This method only considers the effect of the major cutting edge and neglect the tool
nose radius and minor cutting edge. The accuracy of the cutting force prediction
model is greatly affected by the increase of feed to the depth of cut ratio.
2.3.2 Mechanics of cutting analysis for turning operations with sharp cornered and nose radius edged tools
Although various studies of turning operations with conventional plane faced lathe
tools have been found from the literature, the tool corner radius could not be ignored
in some cutting conditions. Investigations were carried out to consider the plane face
lathe tools with sharp or corner radii in high feed to depth of cut rate (f/d) cutting
conditions.
Mechanics of cutting analysis of turning operation with corner lathe tools to allow
for the effects tool tip corner and the secondary cutting edge have only been
considered recently, especially at a high f/d ratios cutting condition. The early
studies in establishment of chip flow direction developed by Stabler [61] and Colwell
[62] have been used and reported [4, 44]. In their studies, the chip flow direction in
turning operation is based on geometrical notions rather than on the mechanics of
cutting analysis. Stabler assumed that the chip flowing direction is in the direction of
resultant chip velocity formed by the vectorial addition of the chip velocities at the
major and minor edge of a sharp cornered conventional lathe tool. The magnitudes of
the velocities at each edge were considered to be proportional to the lengths of the
respective active cutting edges. The directions of the velocities followed the chip
flow rule at each edge. Although the graph showed a comparison of the predicted and
measured chip flow directions, there is no mathematical model in the study. Colwell
22
[62] assumed the chip flow direction in the tool reference plane (Pr) and the tool rake
face to be perpendicular to a line AB, which joins the extreme points of the active
cutting edge or edges by using a sharp cornered and circular profiled lathe tools with
a zero normal rake angle and inclination angle. A reasonable correlation existed
between the predicted and measured chip flow directions for the zero rake and
inclination angle from the results of the experiments. Colwell also attempted to
extend his geometrical concept to lathe tools with non-zero rake and inclination
angles by introducing an approximate ‘correction factor’. Colwell assumed chip flow
to be in the direction of friction face in the rake face. The attempt to use an empirical
force component equation on the elemental basis to determine the resultant friction
force direction and hence the chip flow direction in the rake face has been proved to
be too complex and the empirical data required was not available. An intuitive
geometrical model was suggested and simpler chip flow angle equations for turning
operation with zero rake and inclination angle tools were developed and
experimentally tested.
When the ‘generalised mechanics of cutting analysis’ was applied to the turning
operation [24, 44], the vector AB (joining the extreme points of the active major and
minor cutting edges) has an orientation to the resultant cutting speed (V). This can be
used to determine whether the cutting process should be considered as an orthogonal
or oblique cutting process. Thus, cutting speed is perpendicular to the vector AB (or
parallel to the generalised normal plane), the turning operation can be considered to
involve a two dimensional deformation process (i.e. an orthogonal cutting process).
As a consequence the chip would flow perpendicular to the vector AB in the rake
face irrespective of whether the lathe tool had a zero normal rake and inclination
angle. In general, it was expected that the resultant cutting velocity V would be
inclined by the generalised inclination angle to the ‘generalised’ normal plane
(normal to the vector AB) so that the cutting process could be considered to be a
three dimensional oblique cutting process. Chip flows in the rake face range from an
acute angle to the generalised normal plane to perpendicular the vector AB. In
general, the Stabler chip flow rule (ηc=i) could be applied to the generalised chip
flow angle and the inclination angle. It was also noted that the location and
orientation of the vector AB depend on both the lathe tool geometry and the feed to
depth of cut ratio f/d. Thus for a small f/d ratio the vector AB approached the straight
23
major cutting edge, hence the turning operation could be modelled as a single edge or
‘classical’ oblique cutting process with respect to the straight major cutting edge [44,
63 1963], showing in Figure 2.2.
A
B
True view of vector AB
Straight major cutting edge
Figure 2.2 Concept of equivalent cutting edge
The vector corresponds to the ‘Colwell line’ or ‘equivalent cutting edge’, quoted by
some researchers on chip flow in turning operation [64], and gives the fundamental
meaning to the vector AB in turning operations by the generalised mechanics of
cutting analysis. It is evident that the chip does not always flow in a direction
perpendicular to vector AB as suggested by Colwell, but in general, it will flow at the
generalised chip flow angle with respect to the perpendicular of the vector at
approximately the generalised inclination angle, i.e. the Stabler’s chip flow rule
applies.
Hu et al. applied Lin’s approximate ‘classical’ oblique cutting analysis to predict the
forces and chip flow angle in turning operation with sharp corner lathe tools by
considering the vector AB as ‘an equivalent cutting edge’ [65]. After the equivalent
cutting edge is located, the equivalent cutting edge angles (inclination angle,
approach angle, rake angle and chip flow angle) were determined for each set of
cutting conditions. By assuming Stabler’s chip flow rule applied with respect to the
equivalent cutting edge, the Lin’s oblique analysis was used to predict the cutting
forces for the turning operations. A good correlation was claimed to exist between
the predicted and the measured forces and chip flow angle for a range of cutting
24
conditions. However, their experimental findings are limited because of the low f/d
ratio used (up to 0.125) and inclination angles of only up to 10 degree.
Young [22] found that none of the proposed methods could predict the chip flow
direction with sufficient accuracy unless empirical corrections were introduced.
Young, Mathew and Oxley [22] attempted to apply a similar approach to predict the
forces and chip flow direction in turning operation with a round corner radius tool.
The lathe tools with zero degree normal rake angle, zero degree inclination angle and
various corner radius values were used in their modelling and tests. The turning
operation was modelled as a series of elemental orthogonal cuts, whose elemental
forces were perpendicular to the elemental cutting edges along the tool’s rounded
corner and the straight cutting edge. Two orthogonal elemental force components
were considered to be proportional to the elemental areas of cut. By resolving the
elemental thrust force components in the feed and radial force directions and
summing the elemental force components over the active curved and straight cutting
edges. The total feed and radial force components for each cut were established from
the magnitudes of the resultant friction force and the directions were also determined
at the same time. By assuming the chip velocity as collinear with the friction force
direction on the tool rake face, the chip flow direction angle was determined from the
force analysis. Based on the force analysis, the chip flow direction was predicted for
the tool with zero rake and inclination angles from the tool and cut geometry
analysis.
In order to apply the orthogonal cutting analysis developed by Oxley to predict the
cutting forces in turning operations, Young et al. [22] introduced the ‘equivalent
cutting edge’, which was considered by Hu. The ‘equivalent cutting edge’ in
Young’s model was different to the vector AB (which joins the extreme points of the
active major and minor cutting edges) as defined by Hu and Armarego. Instead
Young et al. defined the ‘equivalent cutting edge’ as the intersection between the
tool rake plane and a plane normal to the predicted chip flow direction. For a zero
rake face plane, the ‘equivalent cutting edge’ is a line normal to the predicted friction
force direction. In addition to this, the ‘modified’ tool angles (normal rake,
inclination and side cutting edge angles) for the ‘equivalent cutting edge’ were
established. The modified normal rake angle and inclination angler were zero degree
25
in his study. The modified major cutting edge angle had to be established for the
rounded corner tools cutting condition. The cutting forces were predicted using
Oxley’s [43, 48, 66] ‘thick zone’ orthogonal cutting analysis with respect to an
equivalent cutting edge. Good correlation between the experimentally measured and
predicted chip flow angle and cutting forces in turning operation with the zero rake
face tools has been reported.
Wang and Mathew [67, 68] have extended Young’s [22] analysis to turning
operation with non-zero rake angle and inclination angle lathe tools. The equivalent
cutting edge in the reference (Pr) plane has been projected from the rake face plane
of a non-zero rake face lathe tool to locate the actual ‘equivalent cutting edge’. The
turning operation was modelled as a single edge ‘classical’ oblique cutting process
by the ‘equivalent cutting edge’ to predict the cutting forces for each set of cutting
conditions. The cutting forces were predicted by the maximum shear stress analysis
preferred by Oxley [69]. The cutting tests were run with tools of inclination angle
and normal rake angle ranged from -10o to +10o and the major cutting edge approach
angles of 10o to 20o for a number of feed to depth of cut (f/d) ratios. The predicted
and measured force components and the chip flow angle were compared and good
agreement has been achieved. Those ‘oblique’ turning analyses rely on the predicted
chip flow direction and the projected ‘equivalent cutting edge’. The ‘equivalent
cutting edge’ is not consistent with the early work by Hu [65] or the vector AB of the
generalised mechanics of cutting analysis proposed by Armarego [44].
The location of the equivalent cutting edge was considered very important in the
study [67, 68]. The tool nose radius and inclinational angle have been found to be
major factors affecting the chip flow direction. The equivalent cutting edge is an
imaginary line in the rake face plane, which is located where it makes an angle with
the straight part of the cutting edge (η0). In order to determine the equivalent cutting
edge, the chip flow angle must be known in advance. The cutting chip can be
considered as a series of elements of infinitesimal width, each having its own
thickness and orientation. The magnitude of the friction force for an arbitrary
element increase or decrease linearly with the local undeformed chip thickness and
its direction is considered to be in line with the local chip direction. The flow of the
entire chip is considered to take the direction of the resultant friction force of the
26
chip. At the same time, it is assumed the local chip flow direction satisfies the
Stabler’s chip flow rule (i.e. the local chip flow angle equal to the local inclination
angle). In calculating the angle of η0, initial consideration is given to the effect of
the nose radius of the cutting tool and then this is followed by the effect of the
inclination angle. In considering the effect of the nose radius, the direction of each
elements friction force is taken to be perpendicular to the local cutting edge. The
magnitude of the friction force acting on an arbitrary chip element can then be found.
The friction force direction (Ω0) can then be determined in this approach. There were
four cases taken into account in exterminating the friction force angle for different
cutting conditions (feed rate, depth of cut, approach angle and the tool nose radius)
[67, 68]. The equivalent cutting edge can be found by the equation:
0'
0 2Ω−−= sC
πη (2.47)
where: η0 (degree) is the angle between the equivalent cutting edge and the straight
part of the cutting edge of the tool, Cs’ (degree) is the value of the approach angle
projected on the rake face, Ω0 (degree) is the chip friction angle.
The modified cutting tool angles (normal rake angle, inclination angle and approach
angle) can be found according to the equivalent cutting edge [67, 68]. The cutting
forces can be predicted with the modified tool angles and cutting conditions. This
approach provides a better solution as it allows for the tool nose radius and
inclination angle to determine the equivalent cutting edge in turning operations.
2.4 Cutting tool wear and its effects on the cutting forces
The above analysis does not consider tool wear cutting conditions. In a practical
cutting process, tool wear is a common phenomenon. It should be studied and then a
better way to predict the cutting force with tool wear cutting condition could be
found.
The cutting forces are decided by practical cutting variables. If tool wears out at the
flank, the friction force will increase and affect the cutting force components
( zyx FFF ,, ). An equation between tool wear and cutting forces should be developed
27
to predict the cutting force. A cutting force model needs to be general and to suit
different cutting conditions. An investigation needs to be performed to determine the
effect of flank wear on the cutting process. If the chip formation process is
independent from wearland, the mechanics of cutting analysis could be used in the
analysis to predict the cutting forces with flank wear. If they are not independent,
which implies the flank wear affects the chip formation process, then the problem
will be more complicated and the classical mechanics of cutting analysis could not be
used to analyse the cutting process. Hence an alternative approach should be
explored.
2.4.1 Wear on cutting tool in machining In all machining process, the tool wears gradually and the work done by the tool
becomes less satisfactory. The roughness on the work surface increases and increases
the cutting forces which lead to intolerable deflections or vibrations. As the tool wear
rate increases, the dimensional tolerances cannot be maintained. Generally, there are
three types of wear on cutting tools: notch wear, wear on the face (crater) and flank
wear, showing in Figure 2.3.
28
VB
Flank WearCrater Wear
Crater
A-A
A A
Figure 2.3 Types of wear on turning tools
Notch wear is a special type of wear which combined flank and face wear and occurs
adjacent to, but outside, the point where the major cutting edge intersects the work
surface. The profile and length of the wear notch depend to a greater extent on the
accuracy of repeated depth settings. Notch wear is excluded from the evaluation of
width of the wearland.
Tool wear on the rake face appears in the form of a small depression called a crater
and hence this form of wear is referred to as crater wear. In many modern cutting
tools, particularly those coated ones, little crater wear occurs in practice. Crater wear
occurs at relatively high cutting speeds and will become negligible compared with the
flank wear at lower speeds [7].
29
The flank wear is the most common type of tool wear. Generally, wearland has a
rather uniform width along the middle portion of the straight part of the major cutting
edge. The width of the wearland is relatively easy to measure. A growing width of the
wearland leads to a reduction in the quality of the tool. The cutting tool normally has
a high initial rate of flank wear which usually decreases considerably after a short
time of use. High-speed steel tools may have prolonged periods of very small
measurable increase of flank wear. This phenomenon occurs especially in low cutting
speeds when machining ductile materials. At high cutting speeds, the increase in flank
wear of all cutting tool materials is usually approximately uniform after an initial high
wear rate. The wearland width (VB) is a suitable parameter to measure tool wear and
to be able to predetermine the value of VB which can be used to assess the tool
condition and is specified by ISO tool-life testing standards [70].
The wearland width is often used to define the end of effective tool life. Once a
certain width level is reached, there will be a major influence on dimensional
accuracy and surface finish of the component as well as the stability of the machining
process [71]. Recently, due to the demand for unmanned machining the intelligent
manufacturing system has grown vigorously. This has produced a need for a reliable
detection technique to predict tool wear in advance.
2.4.2 Effects of tool flank wear on the cutting forces
Although some work has been done to analyse the effect of flank wear on cutting
force and cutting process, the friction condition between the tool flank and the
machined surface is still not well understood. Local friction has some similarities
with the interaction of the tool and chip on the rake face. As noted in [72], with the
same materials, the sliding velocities are of the same order of magnitude as are the
tool rubs against the surface. The difference is that the chip material is highly strained
and the material on the finished surface is comparatively less strained.
In early studies by Kobayashi and Thomsen [73] and Thomsen et al. [74], the friction
on the worn flank of cutting tools has been experimentally studied. They found that
plastic flow occurs in the work-piece beneath the tool flank. It was also found that the
deformation in the primary shear zone and the friction condition on the rake face
30
remain largely unaffected by changes at the tool flank. Okashi and Sata have also
studied the friction phenomenon on the flank of a cutting tool. A linear relationship of
cutting force components with the width of flank wear has been observed [75]. Zorev
[49] also noted this linear relationship. But there is no further analysis on the
phenomenon observed.
Thomsen et al. [74] found that when the tool flank is parallel to the cutting direction,
the cutting force components are virtually constant and do not vary with the size of
the flank wear in orthogonal cutting. Since it is known that the force components
measured during machining varied with flank wear, the flank cannot be parallel to the
cutting direction. So a non-zero inclination angle of the wearland with respect to the
cutting direction is assumed and the inclination angle is determined as a part of the
problem solution.
Zorev [49] showed that forces acting on the tool rake face are responsible for the chip
formation process and the forces acting on the flank do not take part in the chip
formation process. The forces on the flank are produced by the elastic reaction of the
layer of machined material lying under the tool. As the forces on the flank do not
depend on the depth of cut, at a depth of cut approaching zero, the force measured
may be considered as the force acting on the flank. It was noted that flank wear has a
strong influence on the cutting forces. At the same time there was no influence on the
forces acting on the tool rake face. This indicated the constancy of the angles of
inclination of the cutting force curves at different amounts of wear on the flank.
Zorev also confirmed that the absence of influence of the depth of cut on the forces
acting on the flank is consistent when there is no built-up edge or secondary shear on
the tool rake face. Later on it will be shown that a built-up edge influences the
conditions of contact on the tool flank. As the depth of cut is reduced the size of the
built-up edge and secondary shear zone decreases, hence the flank contact conditions
change and there is a corresponding change in the forces acting on it. It is obvious
that in conditions like these the extrapolation method cannot give satisfactory results.
The method of comparing the cutting forces at different amounts of wear on the flank
is based on the following propositions. If all the cutting conditions remain constant
31
and only the width of the wearland increase, the chip formation process does not
generally change. The force acting on the tool rake face therefore does not change
either. But, due to the increase in the area of wearland contact, the force on the flank
increases; the total cutting force also increases correspondingly. The observed
increase in total force must therefore be attributed to an increase in the force acting on
the flank. In other words the increase in the projections of cutting forces is equal to
the increases in the corresponding projections of the forces acting on the flank. The
increases in the projections of cutting forces can be calculated as the different
dynamometer readings at two different values of wearland width. Thus the increases
in the projections of the forces on the flank surface can be determined.
Knowing the increase in the wearland, it is possible to determine the actual forces
acting on the flank. This can be done simply when there is a proportional or
approximately proportional relationship between the wearland width and the force on
the clearance.
Zorev’s study was based on the assumption that change in the wearland width did not
cause a change in the chip length ratio or shear angle. The constant chip length ratio
or shear angle must therefore be monitored when using this method. If it is not
constant, other methods must be used or introduced a correction coefficient for the
cutting forces, which takes into account the change in cutting ratio or shear angle. The
weakness of Zorev’s research is that there is no confirmation of the chip length ratio
or shear angle remaining unchanged with flank wear. The solution is only monitoring
the shear angle.
Stern and Pellini [76] proposed a technique to study the effects of tool wear on
machining forces by isolating flank wear from crater wear. They simulated the wear
through grindings on the flank and the machining forces were plotted as a function of
wearland, crater wear and combined flank-crater wear. They concluded that crater
wear is the dominant wear pattern affecting the machining forces, whereas flank wear
exerts minimal influence and flank-crater wear on the other hand produces a similar
machining force pattern to the crater wear. This finding is totally contradictory to
other studies by Armarego and Brown 1969 [4]. They also found that contrary to
previous studies, feed force may not be a good indicator of flank wear and the
32
assumption that machining force patterns follow the tool life curve is not
substantiated.
Roth and Oxley [77] have shown that for most ductile materials, the velocities of
material particles change gradually in the plastic region. As a result, although there
are exceptions, the flow lines are usually smooth curves [78].
Oxley et al. [79-91] has studied the friction between hard and soft materials and
established a ‘Slip-Line Field Model’ to explain the processes of abrasion, wear and
polishing.
In the basic model [79], the plane strain, perfect-rigid, plastic slip-line fields used by
Challen and Oxley [80-82] to model asperity interactions when a hard surface slides
over a relatively soft one are given in Figure 2.4.
Figure 2.4 Slipline fields: (a) wave model; (b) wave removal model; (c) chip formation model [43]
33
Each of these fields is defined by the slope of the hard asperity α, which can be
looked upon as a surface roughness parameter, and a lubrication parameter f defined
as f=τ/k, with τ the shear strength of the film at the interface DE and k the shear flow
stress of the deforming material. The first model (a), which is for a relatively smooth
well lubricated surface (small α and f values), represents the frictional force as the
force needed to push waves of plastically deformed material along the soft surface
ahead of the hard asperities. The second model (b), which is for rougher surfaces
and/or poorer lubrication, considers the case where the wave is no longer the pusher
ahead of the hard asperity but is torn off (removed). The third model (c), describes
chip formation processes as in machining or, more appropriately in the present
context, as in abrasion.
The coefficients of friction associated with the three models can be determined from
the following equations, which were derived by Challen and Oxley from stress
analysis of their slip-line fields
It can be seen from their study [80, 81, 92] that all the three models only depend on α
and f. They have no dependence on normal force or area of contact. Therefore they
give results in agreement with the basic laws of friction. The three equations show
that µ is predicted to increase with the increase in α. However, a decrease in µ can
occur with an increase in α if the process changes from wave formation (or removal)
to chip formation as a result of the increase α. They also show that an improvement in
lubrication (decrease in f ) results in a decrease in µ in the wave formation range
while giving an increase in µ in the chip formation range.
Waldorf et al. [93] presented a slip-line model of the ploughing process in orthogonal
cutting. The model is based on the assumption of a stable build-up adhered to the
finite radius cutting edge. Instead of material separating at a stagnation point on the
edge, the flow is diverted at the extreme edge of the build-up. It is anticipated that the
model will serve as a starting point for the development of a predictive algorithm for
ploughing forces allowing for simple decomposition of measured forces into
components due to shearing and ploughing. The ploughing process is complex
because it involves both a sliding/indenting action of the tool edge on the workpiece
34
and an interaction with the primary plastic deformation zone associated with the bulk
shearing of the workpiece. The proposed slip-line field representing ploughing
derives parity from previous fields developed for wedge sliding and chip cutting
mechanisms. Unfortunately, they did not study the effect of the flank wear on the
cutting process.
There have been some investigations on the effect of tool wear on the basic cutting
quantities and the cutting performance in orthogonal cutting as well as practical
machining operations. Lin and Pan [94] stated that the understanding of the effect of
tool flank wear on the machining process and machined work-piece is necessary for
improved dimensional accuracy and surface integrity of the component. They
attempted to obtain a full understanding of this complex behaviour by simulating an
incrementally advancing cutting tool from an initial stage of tool-workpiece
engagement to a steady state of chip formation simulation with tool flank wear. They
found that the calculated cutting force and friction force in the tool flank were in
agreement with experiment results. The theoretical frictional force on the flank face
was obtained based on the assumption that the shear angle, contact and sticking
lengths did not apparently change when different lengths of flank wear occur. The
comparison for the cutting forces provided some assurance that the proposed model
and the developed numerical method for orthogonal cutting with tool flank wear are
reasonable and can be extended to study the processes of chip formation during
cutting. They also indicated that based on the strain energy density, which was used
as a chip separation criterion, and twin node processing, the simulation of orthogonal
cutting could be properly undertaken. The separation criteria based on the strain
energy density, was a material constant representing the energy absorption capability
of the workpiece regardless of the cutting conditions. Therefore, the proposed
criterion for chip separation was also acceptable to simulate the cutting process with
tool flank wear. A maximum strain energy density curve had been constructed by
connecting the respective maximum values on each flow line. This curve was found
to be located just within the primary deformation zone and to lie almost parallel to the
shear plane.
Shi and Ramalingam [95] stated that plastic flow occurs in the work-piece beneath
the tool flank and cutting force components measured during machining increase
35
linearly with flank width. They also found that the primary shear zone and friction
condition on the rake face remain unaffected by tool flank wear. They assumed
material under the flank face of the tool to be plastic, and that shear and normal
stresses at the wearland are constant. This assumption is not verified by experiments.
They stated that the wearland produced in orthogonal cutting is not parallel to the
machine surface as is commonly assumed.
Kobayashi and Thomsen [73] found that when the flank face is parallel to the cutting
direction, the cutting force components are virtually constant and do not vary with the
size of the wearland. However, it is well known that force components increase
linearly with the wearland. Shi and Ramalingam [95] stated that due to this
contradiction, the flank face could not be parallel to the cutting direction. Apparently,
this argument exists between different studies.
Based on the above assumptions [43, 77, 78, 92, 96], Shi and Ramalingam [95] used a
slip-line field method to study orthogonal cutting with tool chip breaker and
wearland. They stated that the important purpose of constructing the slip-line field
was to determine the relationship between cutting force components and flank wear.
According to their study, an increase in cutting force components was only caused by
flank wear. The linear increase in cutting force components were observed in the
model and are in general agreement with most observations from previous research
experiments. From their results, they concluded that the cutting force component is a
linear function of flank wear and the normal force component increases more rapidly
than the cutting force component. Their results also show that non-zero strains occur
at and below the machined surface when machining with a worn tool. Severity and
depth of deformation below the machined surface increases with increasing flank
wear. They then suggested that monitoring the ratio of cutting force components can
be a feasible means of tool condition monitoring for flank wear in an automated
machining system. Using this ratio for tool condition sensing may offer a higher
reliability in tool condition sensing when the work materials mechanical properties
are allowed to vary.
Lin et al. [94] studied orthogonal cutting with artificial tool flank wear. The
theoretical force on the flank was obtained from the assumption that the shear angle
36
and the contact and sticking lengths do not apparently change when different lengths
of flank wear occur.
Zhang et al. [97] proposed a three-zone model for orthogonal machining using a
cutting tool with negative flank land. He suggested that there is a ‘Dead Metal Zone’
at the edge of the cutting tool. Compared with the cutting model [41, 42, 66] for a tool
with a sharp edge, the main improvement of model is that it contains three
deformation zones in their model. Shear angles for a sharp tool and a tool with
negative primary land were calculated and verified. It can be noted that when a tool
with negative primary land is used, the shear angle is 20-30 less than that of using the
sharp tool under the same cutting conditions. (Feed rate=0.23mm/rev; Rake angle=00;
Negative primary land Rank angle = -300; Length of Negative primary land=0.2mm)
and (Feed of rate=0.39mm/rev; Rake angle=00; Negative primary land Rank angle = -
300; Length of Negative primary land=0.2mm). With the increase in cutting speed and
the decrease of the feed rate, the influence of the negative primary land on shear angle
increases. Unfortunately, he did not consider the effects of tool edge radius and tool
flank wear.
Recently, it is interesting to note that X. Luo et al. [8] conducted a work of modelling
flank wear in metal cutting. The thin shear zone analysis for orthogonal cutting is
used in their study. But the orthogonal analysis is directly used in turning operation
without considering the effects of tool nose radius and edge force. That will result in a
concept error in modelling of cutting forces in turning operation.
2.4.2.1 Classical Taylor’s equation
Since there is limited machining theory available for a practical machining situation,
to successfully predict tool wear and tool life, one is compelled to rely on Taylor’s
empirical equations [98].
Hung et al. [99, 100] proposed an analytical model for cumulative wear on facing and
turning tools based on Taylor’s classical equation. They assumed that abrasion is the
major mechanism causing tool wear. While assuming constant depth of cut and feed
37
rate, he developed formulas to predict tool life, and to calculate the ‘equivalent’
cutting speed for comparison purposes.
Recently, Choudhury, S.K. et al. [101] proposed a model for evaluating Taylor’s tool-
life constants for a HSS tool on a steel workpiece. Adhesive wear was assumed as the
predominant wear mechanism. But, there were no mechanics of cutting analyses in
the studies.
Oraby et al. [102] carried out a tool wear and force experiment. It has been found that
initially, when a new tool is used, its edge is quickly broken and a finite initial
wearland and initial force value is developed. The amount of the initial wear is
dependent on the cutting conditions but the values of wear depend on different
locations: nose, flank and notch, are generally similar. Following this primary stage,
the tool progressively wears at an almost constant rate until it reaches the final stage
during which wear develops at an increasing rate leading to tool failure. During the
second stage, tool wear takes place at rates which demand that the time required for a
tool to attain a certain wear level, or to reach its final stage of life span, is lower for
more severe conditions.
Force-time curves indicate a similar trend to that of the gradual wear-time
relationship. After a few seconds of cut, a fixed value for the initial force is
established followed by a second phase at which a longer period of uniform force is
noticed. Finally, the forces exhibit higher rates of change near the end of the tool’s
life. The more severe these conditions are the higher the wear rates, which will result
in greater variation in force values. In the third region of the force-time
characteristics, the forces vary differently for various force components. The radial
component of force is usually the one most affected by the wear level at the nose
area.
A quantitative explanation of the characteristic of wear, force and their interrelation
may be obtained by a mathematical formulation of the relevant parameters based on
Taylor’s equation. The nonlinear regression procedure has resulted in a wearland
model [102].
38
A similar study was carried out by Ravindra at el. [103]. He proposed a ‘multiple
regression analysis’ model to describe the wear-time and wear-force relationships for
turning operations. Cutting force components have been found to correlate well with
progressive wear and tool failure. The results show that the ratio between force
components is a better indicator of the wear process when compared with the estimate
obtained by using absolute values for the forces.
It has been observed that the wear curves have clearly defined regions of running in,
steady state and rapid wear. The increasing trend of wear propagation with cutting
speed and occurrence of inflexion in wear characteristics at critical speeds indicate
that wear mechanisms are predominantly thermally controlled. There is an increase in
all components of force with progressive increase in wear. It indicates a good
correlation between wearland VB, feed force and radial force.
Recently, Oraby and Hayhurst [36] conducted an experiment to investigate
relationships between tool life and tool force ratio variation. Non-linear regression
analysis techniques have been used to establish models for wear, tool life, and initial
cutting conditions in terms of forces rather than absolute values of force. It has been
shown that the thrust force provide a sensitive measure of nose, flank and notch wear.
The empirical equations involve a number of constants that are not readily available.
Furthermore these constants depend on many factors, thus requiring a huge amount of
data for a general workshop situation. To obtain and manage such a huge amount of
data is an extremely difficult task. This is one of the main problems encountered in
determining optimum cutting conditions for a process such as turning. Hence there is
no systematic way to quantitatively describe the influence of cutting conditions, tool
geometrical parameters, work and tool material properties, etc. on tool wear and life.
2.4.2.2 Tool flank wear in milling machining
There are several studies on milling machining and tool wear. This is assumed to be
similar to an oblique cutting or turning operation.
39
It has been proposed [104] that the effect on the forces due to flank wear on the
inserts is modelled by adding two force components in face milling, a force normal to
the wearland, and another due to friction on the wearland. The components [105] are
proportional to the material hardness (Brinell hardness number (BHN)), the wearland
width (VB), the wearland length and the coefficient of sliding friction between the
workpiece and tool materials.
The wearland is the contact length between the tool flank and the fresh cut workpiece
surface. It is assumed to be equal to the uncut chip length (width). When cutting
occurs on the tool radius and side cutting edge, the chip length (width) is made up of
two components, one along the nose radius and one along the side cutting edge. The
modified force equations for the radial, longitudinal and tangential force components
including the effects of wear are found by adding appropriate components of the wear
forces to the non-wear forces of equations.
Teitenberg et al. [16] indicated that there is good correlation between the flank
pressure parameter and tool wear in milling. The flank pressure parameter is seen to
be very sensitive to the level of tool wear whereas the flank friction parameter is less
sensitive to tool wear. Results of experiments conducted showed that the spindle
speed had a strong effect on flank pressure parameter but no effect on the flank
friction parameter. The results also showed a weak relationship between the flank
process parameters and the feed rate. In addition, there are consistent and reasonable
relationships between the cutting conditions and the flank process parameters. He
proposed that as the tool flank wear increased, the forces increased owing to the
growth of the tool flank-workpiece interaction. Therefore, it is assumed that the
forces acting on the tool can be divided into two components: rake force and flank
force. Thus flank force can be obtained by subtracting the sharp tool force from the
total force acting on a worn tool. When tested, similar study was carried out and
empirical equations were proposed by Bayoumi et al. [106]
It also was found by Armarego et al. [107, 108] that the introduction of the tool wear
results in a significant increase in the cutting forces while the increased components
can be very well treated as additional edge type forces representing the ‘rubbing’ and
‘ploughing’ forces on the flank surface of the cutting tool in milling.
40
The use of cutting force and acoustic emission (AE) has been applied to the detection
of changes in milling tool insert geometry (tool wear) by Wilcox et al. [109]. The
primary objective of their study was to draw quantitative conclusions as to how local
insert geometry at the cutting edge affects the cutting force and AE. The cutting force
and AE measurements have been made for ranges of artificial wear geometries. The
experiment results showed that force component seemed to be more sensitive to the
change in flank wear. AE was more sensitive to flank wear too, but there was a slight
variation in AE with flank wear. He concluded that flank wear can be expected to
produce an increase in both cutting force and acoustics. Any attempt to relate cutting
force and acoustic emission signals to natural wear requires a detailed knowledge of
the shape of the wearland and its effects on wear progresses.
It is well known that tool wear affects the cutting dynamics during a machining
operation, which in turn alters the cutting forces. In general, the cutting forces will
increase along with the tool wear increase. The cutting force is an important
parameter in providing the flank wear information.
2.5 Concluding remarks
Although considerable fundamental research on the mechanics of cutting in the
conventional material removal process for a sharp tool has been carried out and
published in the literature, the geometrically simple orthogonal cutting process has
been devised to study the cutting mechanics and development of predictive models.
A more general case of a single edge wedge tool cutting referred to as ‘classical’
oblique cutting has also been studied. This oblique cutting process has been thought
to be more representative in a number of practical machining processes, such as
turning, drilling and milling. A variety of mechanics of cutting analysis or models
have been reported in the literature which relate the forces and power in orthogonal
cutting to the deformation process parameters depicted by the shear angle φ, friction
angle β and shear stress τ at the rake face for a particular tool and workpiece material
combination, with practical cutting variables, such as the cut thickness t, width of cut
b, cutting speed V and tool rake angle γ. Those models used for predicting the cutting
41
process were found to be not as simple as originally thought due to the observed high
shear stress of conventional material on sliding friction tests. These were considered
to be a result of the very high shear strains, strain rates and temperatures in the shear
zone and tool-chip interface and also the high normal loads and velocities during
machining.
Furthermore the various proposed relationships between the shearing and friction
process expressed as ‘shear angle relationships’ have been found to be material
dependent. Experimental evidence shows the existence of concentrated ‘edge forces’
at tool cutting edge due to ‘rubbing’ or ‘ploughing’ in the chip deformation process
in 1980’s. The cutting force models must be modified to allow for the edge forces
and to better represent the observed phenomena and experimental trends. The edge
forces were unable to be determined accurately from separate rubbing or ploughing
analysis, but had to be found experimentally.
It appears that the effect of flank wear on the cutting process and its performance still
need further study. In the early study of flank wear, the tool was considered to be
perfectly sharp. This is the fundamental error in those studies. Although many
researchers have developed methods to predict tool wear, they only set up cutting
force and tool wear relationship function by experimenting in a specific cutting
condition to forecast tool wear. Before 1980’s, the limitation of the study on the effect
of flank wear on the cutting performance ignored the edge forces. This resulted in
unacceptable results. Also there has been no study reported on the development of
comprehensive force predictive models which allow for tool wear effect. Currently,
there are no general cutting force models and no shear zone analysis model research
has been carried out which allows for tool flank wear.
The project is based on experimental investigations in orthogonal cutting tests. The
thin shear zone analysis is reviewed and applied to allow for the effects of edge force
and flank wear. An orthogonal cutting force model will be proposed which allows for
the flank wear. The orthogonal cutting database will be developed to predict the
cutting force in orthogonal cutting and will be applied in oblique cutting and turning
operations. A new flank wear force model will also be proposed. The next step will
be to study oblique cutting by using the thin shear zone analysis which allows for
42
flank wear. The oblique cutting force models will be established. The oblique cutting
force models will be further extended to general turning operations. The equivalent
cutting edge will be used to allow for the effect of tool nose radius. Based on the
equivalent cutting edge and modified cutting tool angles, the thin shear zone analysis
for oblique cutting will be applied to turning operations to predict the cutting forces
under flank wear cutting conditions. The cutting tests for oblique cutting and turning
operations will be carried out to verify the cutting force models.
43
Chapter 3 Mechanics of cutting analysis for orthogonal cutting allowing for tool flank wear
effects In fundamental machining investigations, primary attention has been paid to
modelling the geometrically simple orthogonal cutting process, which only involves
two dimensional plastic deformations. Orthogonal cutting tests have been used to
understand the basic cutting process and obtain basic cutting data for use in the
analyses of oblique cutting and other practical machining operations. Although
practical machining operations use more geometrically complex cutting tools than
the wedge tools used in orthogonal cutting, the basic material removal process is
always the same [69]. By using the mechanics of cutting analysis, the orthogonal
cutting process has been related to the oblique cutting process [5, 39]. The
orthogonal and oblique cutting process can be eventually related to practical
operations such as turning and milling for the prediction of cutting performance.
Tool wear is a common phenomenon in machining, a cutting tool gradually wears
with cutting time on the flank (flank wear) and rake face (crater wear). It has been
reported [49, 108] that a tool with crater wear behaves like a restricted contact tool
and, as such, the crater wear on the rake face results in a slight decrease in cutting
forces. Furthermore, crater wear occurs at relatively high cutting speeds and becomes
negligible compared with the flank wear at low speeds. With modern high wear
resistant and coated cutting tools, little crater wear occurs in practice comparing with
that in flank wear. As a result, it is believed [108] that the large ploughing or rubbing
force on the wearland at the tool flank is responsible for the increase in total cutting
force and power and in turn changes the domain from which the economic or
optimum cutting conditions are selected. Thus, the effects of flank wear will be
considered in developing the predictive force models in this thesis.
In the last decades, several important investigations have been carried out on the
effect of tool wear on the basic cutting or shearing process involved in machining
[15, 31, 95, 106, 110]. However, there is a general lack of study on the effect of tool
wear on the basic cutting process as represented by the basic cutting quantities, and
44
no fundamental model is available for predicting the cutting forces and power in
machining for a tool with flank wear. Thus, there is a distinct need to study the
mechanics of cutting and develop cutting force models to incorporate tool wear
effect. This task requires an enormous amount of fundamental analysis of the chip
formation process similar to the work for ‘sharp’ tools over the last 60 years or so.
Whether or not the early work for ‘sharp’ tools can be used to develop cutting force
models for tools with flank wear remains to be fully investigated. One possible
approach is to study the effect of tool wear on the basic cutting process as
represented by the basic cutting quantities, i.e. the shear angle and shear stress in the
shear plane and friction angle in the tool-chip interface. If the tool flank wear can be
proven to have no effect on the basic cutting (shearing) process, a new mechanics of
cutting model can be established based on the early work for ‘sharp’ tools and taking
into account the ploughing or rubbing process on the wearland. Otherwise, a study
similar to that for ‘sharp tool’ cutting may need to be started at an initial stage.
To establish equations or models for quantitatively predicting the cutting
performances such as the cutting forces and power, both empirical and fundamental
approaches may be used. In the empirical approach, experimentally measured
machining characteristic values such as the forces and tool-life are related to the
cutting conditions by regression analysis. This approach involves considerable
testing to determine the constants in the empirical equations and the results were only
applied to the machining operations tested. Given the significantly large and
unmanageable number of tool-work material combinations, cutting and tools
variables and different practical machining operations, the empirical approach is
clearly undesirable in practice.
In order to establish predictive cutting performance models, a mechanics of cutting
analysis for classical orthogonal cutting with tool flank wear is presented in this
chapter. To study the effects of the tool flank wear on the basic cutting properties
(chip length ratio, shear angle, friction angle and shear stress) and cutting forces, a
series of orthogonal cutting tests was carried out under a wide range of tool
geometrical specifications (tool rake angle, wearland) and cutting conditions (cutting
speed, cut thickness) on a free machining steel. Data obtained directly and indirectly
from experiments were analysed qualitatively and quantitatively to explore the trends
45
of the measured force components, deformation and basic cutting quantities with the
variation of the major cutting variables. Comparisons were also made between the
cutting quantities when orthogonal cutting with ‘sharp’ tools and those with flank
wear. The thin shear zone orthogonal cutting analysis was reconsidered in this study.
From this analysis, a modelling approach was proposed to develop cutting force
models when tool flank wear is present. The basic cutting quantities as well as the
force intensities of the cutting edge (edge force) and wearland were also developed
by statistically analysing the experimental data from the orthogonal cutting tests.
3.1 Orthogonal cutting tests The experimental design for orthogonal cutting tests, the associated equipment used,
and data acquisition procedures to meet the objectives of this investigation will be
discussed in this section.
The orthogonal cutting tests were conducted by using both ‘sharp’ tools and tools
with flank wear. The experiment was conducted on a Leadwell CNC lathe cutting a
mild carbon steel, CS1020, whose major chemical compositions and mechanical
properties are given in Table 3.1. The cutting tools used were grade TP20 carbide
flat-top inserts with 8 µm TiN coating. The carbide insert is supplied by SECO
(TPUN160308, TP20).
Table 3.1 Chemical compositions and mechanical properties of CS1020 steel
C Mn P S Density ρ (kg/m3)
Tensile strength (MPa)
Hardness (BHN)
0.2% 0.6% 0.06% 0.06% 7.840x103 380 120
The workpieces were prepared in tubular shape with a wall thickness of 3 mm and
diameter of 100mm, and were machined from an end as shown in Figure 3.1. The
feed rate set by the lathe corresponds to the cut thickness t in orthogonal cutting and
the wall thickness of the workpiece represents the width of the cut b.
46
Figure 3.1 CNC lathe, workpiece and tool set-up in orthogonal cutting tests
A range of cut thickness, rake angle and cutting speed have been selected for the
tests. Specifically, four levels of the cut thickness (0.1, 0.17, 0.24 and 0.31 mm) were
tested at three levels of cutting speed (100, 150 and 200 m/min) and three levels of
tool rake angle (-5°, 0° and 5°). The selection of the range of cutting conditions is
based on the International Standard of Tool-life testing with single-point turning
tools and the Leadwell CNC lathe machining ability. In addition, four levels of
wearland size (0, 0.2, 0.4 and 0.6 mm) were selected, where wearland size ‘0’ mm
stands for sharp tools. The wearland sizes were selected according to the
International Standard ISO3685 [70] and some were higher than the recommended
value in order to study the force pattern for tool condition monitoring in future
investigations. The wearland was artificially generated on the cutting tools by a
lapping process and checked frequently under a shadowgraph for its size. Care was
taken to make sure that the final artificial wearland was within 3% of its specified
size. The specified sizes were used in qualitative analysis but the actual sizes were
used in the regression analysis for the basic cutting quantity database.
Full factorial experimental design was used in the current study, thus a total of 144
tests with three specially made tool holders (one for each rake angle) and 12 inserts
(3 rake angles x 4 wearland sizes) were conducted. Table 3.2 summarises the levels
of parameters used in the whole study.
47
Although Taguchi approach would have higher efficiency in an experimental design,
the prime motivation behind the Taguchi experiment design technique is to achieve a
reduced variation. This technique, therefore, is focused to attain the desired quality
objectives in all steps. The full factorial experimental design does not specifically
address quality. Also the Taguchi methods predicted the combination of factors that
results in the lowest survival of the malignant cells. Taguchi methods agreed with the
conclusions of the full experimental design in this work. However, it must be
stressed that the Taguchi methods are not intended to be a replacement for traditional
experimental design, but is used as a complimentary strategy for analysis of complex
interactions of feasible and practical problems. Taguchi methods are useful for
complex systems [111]. Taguchi methods are more sensitive to the experimental
parameters and data. If an experimental data is wrongly classified in Taguchi Arrays,
the total results were not correct. For these reasons, the Taguchi approach is not used
in the experimental design.
For a maximum cut thickness of 0.31mm (feed rate), the resultant cutting speed angle
was only 0.057o. This could cause a maximum difference of 0.00005% between the
resultant cutting speed Vw and the tangential speed V. Therefore, the resultant cutting
velocity Vw might be considered to be perpendicular to the cutting edge to ensure that
the cutting process is orthogonal.
Table 3.2 Parameters and levels used in orthogonal cutting tests
Wearland VB (mm) 0 0.2 0.4 0.6
Cut thickness t (mm) 0.1 0.17 0.24 0.31
Rake angle γ (degree) -5° 0° 5°
Cutting speed V (m/min) 100 150 200
The computer data sampling system used in the cutting tests to collect the cutting
force data from the dynamometer runs on the Microsoft Windows operation system,
with a data acquisition card (A/D) from the Computer Boards inc. (DAS08-AOH) and
a three-channel data logger program. The data-sampling program was used to allow
the user to choose different scaling factors for the inputs voltage for the amplifier,
thus allowing easy calibration. This system setup can measure up to +10V signals
48
from the Kistler type 5001 charge amplifiers, and the converted digital data is stored
in ASCII format for further processing.
Tool
Kistler 9257A Dynamometer
Kistler 5001Charge Amplifiers
Computer A/D Sampling System
Data recording
Data processing
Figure 3.2 Diagram of cutting force measuring system
As showing in Figure 3.2, the cutting and thrust force components were measured
using a Kistler type 9257A three-component piezoelectric dynamometer which was
mounted on the tool post with a specially made rest. The cutting tool was held on the
top of the dynamometer. The induced power and thrust force signals were processed
and amplified by two Kistler type 5001 charge amplifiers. The amplified signals
were then recorded for further processing by a computer through an A/D converter
and in-house developed data acquisition software. In the current experiment, the
computer was set to log data at 50 samples per second. The final results were taken
from the average of 20 force samples in the steady cutting stage.
For evaluating the chip length ratio (rl) and shear angle (φ), chip samples with the
head and tail of the chip being trimmed out were collected. The weight and length of
chips were measured with precision scale and instrument. At least threes
49
measurement were taken for each quantity and the average was taken as the final
reading. The experimental data is listed in Appendix A.
3.2 Data processing and evaluation procedure
Two measured cutting force components, power force Fcm and thrust force Ftm, as
well as chip length lc and chip weight Wc for each cut were considered as directly
measured quantities. Other cutting quantities such as the chip length ratio rl, shear
angle φ, friction angle β, and shear stress τ can be derived from the above directly
measured data.
Each measured force component from the experiment can be obtained based on the
average of twenty readings. The twenty readings must be chosen at the steady state
of the cutting operation as shown in Figure 3.3. The readings taken from an unsteady
state will result in inaccurate date which in turn affects the analysis.
Figure 3.3 Data sampling at steady state area
One of the indirect cutting properties, chip length ratio rl, can be obtained using the
measured chip length lc and chip weight Wc by considering the continuity and
incompressibility condition of the general theory of plasticity. The assumption for
this theory is that the density and the volume of the material are the same before and
after the chip formation process. The following equations have been used to calculate
the chip length ratio rl:
Wc=ρbtl=ρbctclc (3.1)
50
rl = lc / l=ρbtlc/Wc (3.2)
where: ρ = Density of work material (g/mm3)
t = Cut thickness (mm)
b = Width of cut (mm)
Wc = Weight of chip (g)
lc = Chip length (mm)
In order to estimate the other cutting quantities (shear angle φ, friction angle β, and
shear stress τ) from the measured quantities, the derived relationships as given in the
previous chapter have been used in the reverse order. To facilitate the analysis, the
equations of the ‘classical’ orthogonal cutting force components and processing
variables are summarised here based on the thin shear zone model [5]:
tKbtF ccs =−+
−=)cos(sin
)cos(γβφφ
γβτ (3.3)
tKbtF tts =−+
−=)cos(sin
)sin(γβφφ
γβτ (3.4)
+== −−
cs
ts
FF11 tantan γµβ (3.5)
τφ φ φ
=−( cos sin ) sinF F
btcs ts (3.6)
−= −
γγφ
sin1cos
tan 1
l
l
rr
(3.7)
From the above equations, it can be seen that the shear angle φ depends on the
cutting chip length ratio rl. The power and thrust forces at the shear zone (Fcs and Fts)
increase lineally with cut thickness t for a given rake angle. The slopes Kc and Kt
depend on tool rake angle γ, shear angle φ, friction angle β, shear stress τ and width
of cut b. The ‘pure’ shear force components (Fcs and Fts) are used for calculating the
friction angle and shear stress as shown in Eqs. 3.5 and 3.6.
A further development to the thin shear zone analysis is the introduction of a
concentrated forces acting at the cutting edge. They are called edge forces. It has
been suggested by many researchers [4-6] that since the cutting edge is not perfectly
51
sharp, a rubbing or ploughing process could occur in the vicinity of the cutting edge
resulting in an edge force in addition to the forces requested in acting chip formation
in the shear zone. This phenomena is manifested by the positive force intercepts
when the measured force versus cut thickness graphs are extrapolated to zero cut
thickness, and is proportional to the engaged cutting edge length [5]. Armarego has
suggested the removal of the edge force from the measured force data when
evaluating the basic cutting quantities using the thin shear zone model.
Consequently, the friction angle and shear stress can be found from the equations
below [5].
−−+== −−
cecm
tetm
FFFF11 tantan γµβ (3.8)
( ) ( )[ ]bt
FFFF tetmcecm φφφτ sinsincos −−−= (3.9)
Likewise, the total cutting forces could be represented by
cscecscec FbCFFF +=+= (3.10)
tstetstet FbCFFF +=+= (3.11)
Where: Cce and Cte (N/mm) are the edge force intensity factors and can only be
obtained from the orthogonal cutting tests at this stage of development.
It is important to note that the above equations for evaluating the basic cutting
quantities have been developed for ‘sharp’ tools. At this stage, it is not clear if the
chip formation process (or the basic cutting process) remains the same when tool
flank wear is present. However, a reasonable argument is that if the cut thickness is
zero, the tool cuts nothing and the cutting force should be zero. Otherwise, the
cutting force is a result of the rubbing process at the cutting edge and wearland based
on the analysis for ‘edge forces’ [5]. Thus, this rubbing or ploughing force
component can be removed from the as-measured forces in a similar way for ‘sharp’
tools, as that suggested by Armarego [5]. A further analysis can be made to study
52
whether the rubbing force in cutting with tool wear is greater than that with a ‘sharp’
tool, so that the rubbing force on the cutting edge and that on the wearland, if it
exists, can be distinguished and further studied. After the rubbing forces are removed
from the measured forces, what remains are the forces associated with the chip
formation process. These forces can be analysed and compared with those from
cutting with ‘sharp’ tools to see if tool flank wear affects the chip formation or basic
cutting process by examining the respective basic cutting quantities. If tool flank
wear affects the basic chip formation process, a mechanics of cutting analysis and
modelling approach will need to be developed. If tool flank wear is found to have no
effect on the basic chip formation process in the cutting zone, it is logical and
realistic to consider that the increase in the as-measured force is a result of the
increase in the rubbing force because of the presence of the tool wear. Consequently,
the analysis and predictive cutting performance models for cutting with tool flank
wear may be developed based on the earlier analysis for ‘sharp’ tools and taking into
consideration of the rubbing or ploughing force on the wearland. It is on this basis
that the analysis of the cutting data for cutting with tool wear is carried out in this
thesis.
3.2.1 Methodologies of data analysis A series of statistical analysis is planned to study the effects of cutting parameters on
the orthogonal cutting process. The qualitative behaviour with respect to the various
cutting variables will be studied and compared with those from classical orthogonal
cutting tests to assess the effects of tool flank wear. The qualitative variation with
respect to the changes of process variables such as the cutting speed, normal rake
angle, cut thickness and wearland size will be statistically investigated.
In the present study, the cutting force components are expected to be dependent on
cut thickness with positive and substantially significant intercepts. The intercepts can
be treated as rubbing forces similar to earlier studies for ‘sharp’ tools [5]. The
increased amount of the rubbing force when tool flank wear is present may be
considered as the rubbing force on the wearland (or called wearland force). In the
analysis for ‘sharp’ tool cutting [5], it has been found that the cutting speed and rake
angle do not have a significant effect on the intercepts or rubbing forces. To confirm
this effect in cutting with tool flank wear and to check the existence of rubbing
53
forces, linear regression of the measured cutting forces with cut thickness has to be
evaluated. Based on a 95% confidence level, it is proposed that a significant positive
intercept would substantiate the existence of the rubbing forces. A positive force
slope is also expected from the cutting force-cut thickness plots. Furthermore, as it is
expected that the rubbing forces are additional force components, analysis of
covariance will be conducted to exam the parallelist and singularity of the regression
lines representing the relationship between measured force and cut thickness.
Statistical analysis will also be carried out to study the effect of flank wear as well as
other cutting variables (cutting speed, rake angle and cut thickness) on the basic
cutting quantities.
In the study, correlation analysis is carried out to check the linear association
between two scale variables. Two kinds of correlation coefficients have been used.
Generally, the Pearson correlation coefficient works well when the variables are
approximately normally distributed and have no outliers. If the data distribution is
unknown (or not in a normal distribution), the Nonparametric Correlations will be
used to determine the correlation coefficient. For example, when the correlation
analysis is used to check the effect of cut thickness on cutting forces, General Linear
Model considering the covariance is used to check the parallelist and singularity of
regression lines. Where other cutting quantities such as shear angle, friction angle
and shear stress are not significantly affected by the cut thickness, Student t-tests are
used to test the effects of operation variables, i.e. rake angle, cutting speed and flank
wear. Details of the test plan are given in Table, 3.3-3.6.
Table 3.3 Measured forces analysis plan
Variable Anticipated Condition Test Required
Linearly dependent on cut thickness
Correlation test Linear regression analysis Significant correlation
Cut thickness t Positive intercept (forces) expected at zero cut thickness
Significance of regression intercepts Significant intercept
Rake angle γ Regression slopes vary with the rake
Correlation test Linear regression analysis Dependent on rake angle
Cutting speed V Regression slopes
independent of speed
Analysis of multiple regression lines for various speed at given rake angle
and VB
Correlation coefficient is not significantly
different from zero
54
Table 3.4 Chip length ratios and shear angles analysis plan
Variable Anticipated Condition Test Required
Cut thickness t Independent of cut thickness
Correlation of rl or φ with t at given speed V and rake
angle
Non-significant correlation
Rake angle γ Increase with rake
angle Analysis of variance Analysis of variance test
fails (means are not equal)
Cutting speed V Independent of cutting speed
Analysis of variance
Analysis of variance or t-test passes (means
are equal)
Table 3.5 Shear stress analysis plan
Variable Anticipated Condition Test Required
Cut thickness t Independent of cut thickness
Correlation of τ with t at given speed V and rake
angle
Non-significant correlation
Rake angle γ Independent of rake
angle Analysis of variance Analysis of variance test fails (means are equal)
Cutting speed V Independent of cutting speed
Analysis of variance
Analysis of variance or t-test passes (means
are equal)
Table 3.6 Friction angle analysis plan
Variable Anticipated Condition Test Required
Cut thickness t Independent of cut thickness
Correlation of β with t at given speed V and rake
angle
Non-significant correlation
Rake angle γ Increase with rake
angle Analysis of variance Analysis of variance test
fails (means are not equal)
Cutting speed V Independent of cutting speed
Analysis of variance
Analysis of variance or t-test passes (means
are equal)
3.2.2 Shear angle relationship and basic cutting quantities database
The shear angle φ and (β-γ) in the classical orthogonal cutting are the experimentally
derived quantities. A shear angle relationship could reduce one unknown cutting
variable required to analyse the orthogonal cutting process. In this study, a regression
analysis will be made to establish the regression equations between the two variables.
The effect of tool flank wear on the shear angle relationship will be analysed.
Correlation and regression analysis will be carried out for the basic cutting quantities
(friction angle, shear stress, shear angle) with respect to the input cutting variables
55
(cutting speed, rake angle, cut thickness, and wearland size). This analysis will arrive
at a database for the basic cutting quantities that are essential for the development of
predictive cutting performance models.
In the following sections, the analysis of the orthogonal cutting process with tool
flank wear will be carried out. The characteristics of the as-measured forces will be
discussed first to assess if they are different from those in cutting with sharp tools.
This will be followed by an analysis of the characteristics of the basic cutting
properties on the shear zone and tool-chip interface, i.e. the chip length ratio or shear
angle, friction angle, shear stress and shear angle relationship, to study if tool flank
wear affects the basic chip formation process. The outcome of these analyses will
form the basis for proposing the appropriate approach to model the cutting
performance in machining with tool flank wear.
3.3 The characteristics of the measured cutting forces The characteristics of the measured forces with changes in cut thickness, tool rake
angle, cutting speed and wearland size have been studied for both ‘sharp’ tools and
those with flank wear. It has been found that the presence of tool flank wear has no
significant effect on the general trends and characteristics of the measured forces
from the cutting tests when compared with those in orthogonal cutting with sharp
tools, although tool wear did increase the magnitude of the cutting forces.
The typical and representative trends for the experimental forces with respect to the
process variables are shown in Figures 3.4 to 3.7. (From the appendix A, for
V=150m/min, γ=0, the measured forces are found in the table for different t and VB.
Data for other Figures and analysis are found in the same way).
Figures 3.4(a) and (b) show the effect of cut thickness on the measured forces at
different wearland size VB. An increase in cut thickness results in an increase in the
measured power and thrust force components. It is found that the measured power
force and thrust force increase linearly with an increase in cut thickness. The flank
wear does not seem to affect the qualitative trends of the cutting forces.
56
Figures 3.5(a) and (b) show the effect of rake angle on the measured forces when
t=0.1mm, V=150m/min and wearland size VB=0 and 0.6mm. An increase in rake
angle results in a decrease in the measured power force in Figure 3.5 (a). A similar
trend can also be observed for thrust force in Figure 3.5 (b). These trends are again
the same as those in cutting with sharp tools from both the current cutting study and
the literature.
The effect of cutting speed V on the cutting forces is shown in Figures 3.6 (a) and
(b). It can be seen from those figures that the effect of cutting speed on the measured
forces is very marginal, although there is a slight decrease of measured force with
cutting speed when VB=0.6mm. However, this decrease is not significant and can be
neglected, as confirmed by the statistical analysis later in the chapter. This trend
applies for both sharp tools where VB=0 and for worn tools with a wearland.
0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Mea
sure
d Fc
m (N
)
VB=0mmVB=0.4mmVB=0.6mm
v=150m/min, γ=0°
0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Mea
sure
d Ft
m (N
) VB=0mmVB=0.4mmVB=0.6mm
v=150m/min, γ=0°
(a) (b)
Figure 3.4 Effect of cut thickness on measured forces.
0
500
1000
1500
2000
2500
3000
-10 -5 0 5 10Rake angle γ (deg)
Mea
sure
d Fc
m(N
)
VB=0.6mmVB=0mm
t=0.1mm, V=150m/min
0
500
1000
1500
2000
2500
3000
-10 -5 0 5 10Rake angle γ (deg)
Mea
sure
d Ft
m(N
)
VB=0.6mm
VB=0mm
t=0.1mm, V=150m/min
(a) (b) Figure 3.5 Effect of rake angle on the measured forces
57
γ=0deg, t=0.1mm
0
500
1000
1500
2000
2500
3000
50 100 150 200 250Cutting speed V (m/min)
Mea
sure
d Fc
m (N
)VB=0.6mmVB=0mm
γ=0deg, t=0.1mm
0
500
1000
1500
2000
2500
3000
50 100 150 200 250Cutting speed V (m/min)
Mea
sure
d Ft
m (N
)
VB=0.6mmVB=0mm
(a) (b) Figure 3.6 Effect of cutting speed on the measured forces
0
500
1000
1500
2000
2500
3000
0 0.2 0.4 0.6Wearland size VB (mm)
Mea
sure
d Fc
m (N
)
γ=−5γ=0γ=5
V=150m/min, t=0.17mm
0
500
1000
1500
2000
2500
3000
0 0.2 0.4 0.6Wearland size VB (mm)
Mea
sure
d Ft
m (N
)γ=−5γ=0γ=5
V=150m/min, t=0.17mm
(a) (b)
Figure 3.7 Effect of wearland size on the measured forces
The effects of the tool wearland on the cutting forces are investigated by analysing
the regression lines of the measured forces Fcm and Ftm as shown in Figure 3.7. The
effect can also be seen from Figure 3.4. It can be noticed from Figure 3.7 that the
measured force components Fcm and Ftm increase linearly with the wearland size.
This increase may be a result of the rubbing or ploughing force on the wearland, or
the change in the basic chip formation process, or both, which have yet to be
determined in this chapter. It can be noticed from Figure 3.4 that under different cut
thicknesses, the measured forces increase linearly with cut thickness and the
regression lines are almost parallel for different wearland sizes, including VB=0 for
sharp tools. It appears that wearland size does not affect the trend of the forces or the
basic cutting chip formation process; but it changes the intercepts of the regression
lines. By extending the trend lines of the measured forces to zero cut thickness,
58
positive intercepts for all the forces can be obtained as shown in Figure 3.4, This
indicates the existence of rubbing or ploughing force on the wearland and the
rubbing force increases with the wearland size.
Table 3.7 Regression and correlation analysis of the measured power force
component Fcm in terms of cut thickness.
VB V γ Correlation Coefficient
Test statistics for intercept
Significance at 95% Confidence Level
R2 t P-value Intercept Correl. Coef 0 100 5 246.033 0.9999 36.009 0.001 Yes Yes 0 0 186.235 0.9999 38.802 0.001 Yes Yes 0 -5 412.172 0.995 14.703 0.005 Yes Yes 0 150 5 188.134 1.000 49.912 0.000 Yes Yes 0 0 183.439 1.000 10.817 0.000 Yes Yes 0 -5 312.267 0.998 23.178 0.002 Yes Yes 0 200 5 248.767 0.999 39.003 0.001 Yes Yes 0 0 250.978 1.000 100.37 0.000 Yes Yes 0 -5 269.562 0.998 22.564 0.002 Yes Yes
0.2 100 5 306.767 0.985 8.161 0.015 No Yes 0.2 0 276.246 0.999 26.510 0.001 Yes Yes 0.2 -5 392.825 0.996 15.715 0.004 Yes Yes 0.2 150 5 297.457 0.998 20.558 0.002 Yes Yes 0.2 0 206.341 0.998 22.545 0.002 No Yes 0.2 -5 394.257 0.994 13.343 0.006 Yes Yes 0.2 200 5 355.385 1.000 54.092 0.000 Yes Yes 0.2 0 298.429 1 106.94 0.000 Yes Yes 0.2 -5 312.476 1 70.818 0.000 Yes Yes 0.4 100 5 399.265 0.985 8.011 0.015 No Yes 0.4 0 134.771 0.996 10.777 0.059 No Yes 0.4 -5 669.449 0.993 12.145 0.007 Yes Yes 0.4 150 5 380.779 1 63.244 0.000 Yes Yes 0.4 0 333.261 1 28.0 0.001 Yes Yes 0.4 -5 650.973 0.992 10.88 0.008 Yes Yes 0.4 200 5 430.188 0.999 36.614 0.001 Yes Yes 0.4 0 598.675 0.997 14.081 0.045 No Yes 0.4 -5 633.483 0.995 13.460 0.005 Yes Yes 0.6 100 5 385.656 0.995 14.518 0.005 No Yes 0.6 0 532.854 0.1 314.22 0.000 Yes Yes 0.6 -5 835.973 0.962 4.991 0.038 No Yes 0.6 150 5 36.325 0.954 4.524 0.046 No Yes 0.6 0 575.692 0.996 15.265 0.004 Yes Yes 0.6 -5 547.928 0.999 28.337 0.001 Yes Yes 0.6 200 5 634.198 0.789 1.814 0.211 No Yes 0.6 0 399.209 0.996 14.952 0.004 Yes Yes 0.6 -5 587.112 0.998 4.311 0.002 Yes Yes
Quantitative analysis of the effects of process variables on the measured cutting
forces is also conducted. The results of the statistical analysis about the effect of the
cut thickness on the measured forces are given in Tables 3.7 and 3.8 for cutting force
and thrust force respectively. The test results show that there are significant
correlations between the measured cutting forces and cut thickness for both ‘sharp’
59
and worn tools. The results confirm the linear relationship between cut thickness and
the measured forces for cutting with both tool flank wear and without tool flank
wear. The tables also show that the linear relationship between the cut thickness and
cutting forces has a significant intercept, confirming the existence of the rubbing
force. This has early been noticed in the literature to increase with an increase in the
wearland size.
Table 3.8 Regression and correlation analysis of the measured thrust force component Ftm in
terms of cut thickness.
VB V γ Correlation coefficient Test statistics for intercept Significance at 95% confidence level
R2 t P-value Intercept Correl. Coef 0 100 5 179.178 0.963 1.346 0.310 No Yes 0 0 93.094 0.981 0.755 0.529 No Yes 0 -5 356.949 0.995 5.500 0.032 Yes Yes 0 150 5 99.948 0.944 0.578 0.622 No No 0 0 247.737 1.000 16.035 0.004 Yes Yes 0 -5 332.094 0.995 6.343 0.033 Yes Yes 0 200 5 316.934 0.983 5.071 0.037 Yes Yes 0 0 365.648 0.999 21.355 0.002 Yes Yes 0 -5 419.787 0.997 12.131 0.007 Yes Yes
0.2 100 5 297.589 0.982 3.911 0.06 No Yes 0.2 0 292.738 0.997 7.379 0.018 Yes Yes 0.2 -5 444.579 0.988 5.054 0.037 Yes Yes 0.2 150 5 283.178 0.995 6.879 0.02 Yes Yes 0.2 0 277.678 0.995 4.523 0.046 Yes Yes 0.2 -5 524.375 0.974 4.953 0.038 Yes Yes 0.2 200 5 384.069 0.974 10.938 0.008 Yes Yes 0.2 0 297.145 0.998 9.915 0.01 Yes Yes 0.2 -5 442.9 0.999 27.976 0.001 Yes Yes 0.4 100 5 442.577 0.975 6.145 0.025 Yes Yes 0.4 0 -32.723 0.998 -0.533 0.658 No Yes 0.4 -5 570.019 0.985 5.444 0.032 Yes Yes 0.4 150 5 397.042 1 50.647 0.000 Yes Yes 0.4 0 387.167 0.996 8.762 0.013 Yes Yes 0.4 -5 699.826 0.953 4.590 0.044 Yes Yes 0.4 200 5 438.549 0.996 17.438 0.003 Yes Yes 0.4 0 945.068 0.970 9.185 0.069 No Yes 0.4 -5 753.081 0.931 5.599 0.03 Yes No 0.6 100 5 402.560 0.992 3.933 0.059 Yes Yes 0.6 0 509.807 0.992 7.148 0.019 Yes Yes 0.6 -5 848.379 0.916 4.476 0.046 Yes No 0.6 150 5 217.051 0.929 0.878 0.473 Yes No 0.6 0 524.191 0.996 13.538 0.005 Yes Yes 0.6 -5 653.380 0.955 7.047 0.02 Yes Yes 0.6 200 5 475.105 0.799 2.91 0.101 No No 0.6 0 459.932 0.983 5.234 0.035 Yes Yes 0.6 -5 586.634 0.996 15.064 0.004 Yes Yes
A regression analysis is also conducted to assess the effect of rake angle on the
measured cutting force components. The analysed results are given in Table 3.9 for
the cutting or power force component Fcm and in Table 3.10 for the thrust component
Ftm.
60
Table 3.9 Regression and correlation analysis of the measured power force component Fcm with rake angle.
V (m/min)
t (mm)
VB (mm) R Square Fcm Slop
Coefficients Standard Error t Stat P-value
Significance at 95%
confidence level
100 0.1 0 0.781905 -13.836 7.307291 -1.89345 0.309335 Yes
100 0.1 0.2 0.950466 -12.5192 2.857999 -4.38041 0.142885 Yes
100 0.1 0.4 0.802653 -32.4666 16.0986 -2.01673 0.293051 Yes
100 0.1 0.6 0.996438 -56.8057 3.396147 -16.7265 0.038015 Yes
100 0.17 0 0.842109 -25.791 11.16769 -2.30943 0.260144 Yes
100 0.17 0.2 0.941428 -12.983 3.238358 -4.00913 0.155617 Yes
100 0.17 0.4 0.821865 -45.417 21.1443 -2.14795 0.277386 Yes
100 0.17 0.6 0.603416 -20.135 16.32342 -1.2335 0.433685 Yes
100 0.24 0 0.990012 -16.629 1.670274 -9.95585 0.063731 Yes
100 0.24 0.2 0.266172 5.996 9.955828 0.60226 0.654901 Yes
100 0.24 0.4 0.997652 -33.562 1.628128 -20.6139 0.030859 Yes
100 0.24 0.6 0.768437 -25.775 14.14912 -1.82167 0.319605 Yes
100 0.31 0 0.971204 -19.551 3.366529 -5.80746 0.108556 Yes
100 0.31 0.2 0.644193 -20.408 15.16699 -1.34555 0.406881 Yes
150 0.1 0 0.982498 -13.2443 1.767673 -7.4925 0.084468 Yes
150 0.1 0.2 0.884894 -6.9063 2.490862 -2.77265 0.220362 Yes
150 0.1 0.4 0.709141 -27.8294 17.82292 -1.56144 0.362632 Yes
150 0.1 0.6 0.98941 -15.112 1.563465 -9.66571 0.06563 Yes
150 0.17 0 0.96503 -24.673 4.696744 -5.25321 0.119754 Yes
150 0.17 0.2 0.89458 -15.954 5.476745 -2.91304 0.210517 Yes
150 0.17 0.4 0.906027 -33.398 10.75604 -3.10505 0.19835 Yes
150 0.17 0.6 0.997551 -22.059 1.092924 -20.1835 0.031516 No
150 0.24 0 0.994541 -20.074 1.487254 -13.4974 0.04708 No
150 0.24 0.2 0.667324 -9.07 6.403969 -1.41631 0.391382 Yes
150 0.24 0.4 0.893974 -47.216 16.26049 -2.90373 0.211144 Yes
150 0.24 0.6 0.988199 -34.458 3.765478 -9.15103 0.069293 Yes
150 0.31 0 0.956116 -23.971 5.135531 -4.66768 0.134358 Yes
150 0.31 0.2 0.080253 -10.16 34.39506 -0.29539 0.817148 Yes
150 0.31 0.4 0.960244 -32.605 6.634332 -4.91459 0.127792 Yes
150 0.31 0.6 0.373925 37.557 48.5973 0.772821 0.581138 Yes
200 0.1 0 0.988133 -11.6907 1.281175 -9.12498 0.069489 Yes
200 0.1 0.2 0.602769 -2.524 2.04897 -1.23184 0.434106 Yes
200 0.1 0.4 0.618669 -22.4633 17.6358 -1.27373 0.423725 Yes
200 0.1 0.6 0.968976 -19.1976 3.435119 -5.58863 0.112721 Yes
200 0.17 0 0.926911 -9.575 2.68872 -3.56117 0.174279 Yes
200 0.17 0.2 0.944864 -5.956 1.438757 -4.13968 0.150894 Yes
200 0.17 0.4 0.545995 -34.407 31.37495 -1.09664 0.470677 Yes
200 0.17 0.6 0.956783 -29.404 6.249239 -4.70521 0.133317 Yes
200 0.24 0 0.965345 -12.277 2.326144 -5.27783 0.119208 Yes
200 0.24 0.2 0.857078 -12.217 4.988884 -2.44884 0.24681 Yes
200 0.24 0.4 0.688125 -41.927 28.22608 -1.4854 0.377213 Yes
200 0.24 0.6 0.008309 -1.258 13.74325 -0.09154 0.941888 Yes
200 0.31 0 0.983258 -25.114 3.27704 -7.66362 0.082604 Yes
200 0.31 0.2 0.66573 -15.669 11.10302 -1.41124 0.392459 Yes
200 0.31 0.6 0.631868 -67.268 51.34491 -1.31012 0.415045 Yes
61
Table 3.10 Regression and correlation analysis of the measured thrust force component Ftm with rake angle.
V (m/min)
t (mm)
VB (mm) R Square Ftm Slop
Coefficients Standard Error t Stat P-value
Significance at 95%
confidence level
100 0.1 0 0.774523 -19.8882 10.73073 -1.85338 0.314991 Yes
100 0.1 0.2 0.999709 -26.5696 0.453047 -58.6465 0.010854 No
100 0.1 0.4 0.552109 -28.2788 25.47038 -1.11026 0.466766 Yes
100 0.1 0.6 0.993048 -58.9786 4.934694 -11.9518 0.053142 Yes
100 0.17 0 0.892896 -41.7262 14.45144 -2.88734 0.212256 Yes
100 0.17 0.2 0.954632 -33.4569 7.29362 -4.58715 0.136645 Yes
100 0.17 0.4 0.857815 -49.6572 20.21681 -2.45623 0.24614 Yes
100 0.17 0.6 0.937086 -29.3516 7.605291 -3.85936 0.161405 Yes
100 0.24 0 0.962242 -46.7545 9.261582 -5.04822 0.124496 Yes
100 0.24 0.2 0.999997 -27.7937 0.047764 -581.895 0.001094 No
100 0.24 0.4 0.976874 -52.8187 8.126834 -6.49929 0.09719 Yes
100 0.24 0.6 0.786517 -44.5039 23.18598 -1.91943 0.305766 Yes
100 0.31 0 0.999351 -40.275 1.026413 -39.2386 0.016221 No
100 0.31 0.2 0.97078 -51.3193 8.903492 -5.76395 0.10936 Yes
150 0.1 0 0.978792 -21.3397 3.141201 -6.79348 0.093042 Yes
150 0.1 0.2 0.999989 -21.4835 0.072359 -296.9 0.002144 No
150 0.1 0.4 0.838662 -33.7788 14.81559 -2.27995 0.26314 Yes
150 0.1 0.6 0.972028 -25.8103 4.378422 -5.89487 0.106977 Yes
150 0.17 0 0.999016 -56.0007 1.757489 -31.864 0.019973 No
150 0.17 0.2 0.9974 -34.1012 1.741184 -19.5851 0.032477 No
150 0.17 0.4 0.960489 -43.3371 8.789719 -4.93043 0.127392 Yes
150 0.17 0.6 0.989515 -37.8938 3.900682 -9.71466 0.065302 Yes
150 0.24 0 0.946169 -32.7442 7.810256 -4.19247 0.149063 Yes
150 0.24 0.2 0.999957 -29.4454 0.192581 -152.899 0.004164 No
150 0.24 0.4 0.930786 -61.1803 16.68332 -3.66715 0.16948 Yes
150 0.24 0.6 0.996273 -54.5222 3.334637 -16.3503 0.038888 No
150 0.31 0 0.962213 -47.5138 9.415775 -5.04619 0.124545 Yes
150 0.31 0.2 0.554361 -27.6437 24.78513 -1.11533 0.465324 Yes
150 0.31 0.4 0.994409 -45.7342 3.429345 -13.3361 0.047647 No
150 0.31 0.6 0.044966 -5.9341 27.34764 -0.21699 0.86397 Yes
200 0.1 0 0.99893 -23.2484 0.760722 -30.5609 0.020824 No
200 0.1 0.2 0.696149 -14.9149 9.853701 -1.51363 0.37168 Yes
200 0.1 0.4 0.327158 -31.0614 44.54503 -0.6973 0.612353 Yes
200 0.1 0.6 0.876088 -26.0288 9.788962 -2.659 0.229004 Yes
200 0.17 0 0.837435 -22.5826 9.949743 -2.26966 0.2642 Yes
200 0.17 0.2 0.897273 -16.6393 5.630066 -2.95543 0.207708 Yes
200 0.17 0.4 0.420112 -41.5124 48.77166 -0.85116 0.551077 Yes
200 0.17 0.6 0.929097 -36.6773 10.1321 -3.61991 0.171587 Yes
200 0.24 0 0.960517 -27.817 5.639806 -4.93226 0.127346 Yes
200 0.24 0.2 0.979581 -26.4568 3.819715 -6.92638 0.091282 Yes
200 0.24 0.4 0.612452 -48.8592 38.86632 -1.25711 0.427793 Yes
200 0.24 0.6 0.993438 -31.3111 2.544666 -12.3046 0.051625 Yes
200 0.31 0 0.957321 -42.985 9.075969 -4.73613 0.132472 Yes
200 0.31 0.2 0.999056 -29.986 0.921913 -32.5258 0.019567 No
200 0.31 0.6 0.690527 -58.5314 39.18405 -1.49375 0.375561 Yes
62
The analysis (Table 3.9 and 3.10) shows that the slopes of regression lines of the
measured forces with respect to rake angle are negative at 95% confidence level.
There are only a few random cases where the slopes are not significant. This result
confirms that an increase in rake angle results in a reduction in the measured cutting
forces components, and that tool flank wear does not qualitatively change the effect
of rake angles on the overall cutting forces.
An analysis of covariance (ANCOVA) was performed to study the effect of cutting
speed on the measured forces. The results are given in Table 3.11.
Table 3.11 Linear regression with covariance analysis of the measured forces components vs
cutting speed V under different cut thickness t.
Power force Fcm
slopes intercepts Parallel Single VB (mm) rake angle F P F P
0 -5 0.39 0.695 1.2 0.365 yes yes 0 10.27 0.012 2.92 0.13 no yes 5 1.88 0.232 1.36 0.326 yes yes
0.2 -5 0.1 0.91 0.47 0.649 yes yes 0 4.59 0.062 1.02 0.415 yes yes 5 0.39 0.693 0.12 0.892 yes yes
0.4 -5 0.16 0.854 0.03 0.969 yes yes 0 2.88 0.168 8.14 0.039 yes no 5 0.68 0.544 0.1 0.909 yes yes
0.6 -5 0.83 0.481 1.39 0.319 yes yes 0 2.27 0.184 1.86 0.236 yes yes 5 1.96 0.221 0.74 0.516 yes yes
Thrust force Ftm
slopes intercepts Parallel Single VB (mm) rake angle F P F P
0 -5 4.08 0.076 0.66 0.55 yes yes 0 6.2 0.035 3.56 0.095 no yes 5 0.84 0.478 0.7 0.532 yes yes
0.2 -5 1.09 0.394 0.34 0.725 yes yes 0 5.44 0.045 0.05 0.952 no yes 5 2.15 0.198 1.02 0.414 yes yes
0.4 -5 1.87 0.234 0.51 0.625 yes yes 0 19.1 0.009 46.2 0.002 no no 5 1.72 0.257 0.32 0.735 yes yes
0.6 -5 0.45 0.658 1.21 0.363 yes yes 0 2.09 0.204 0.24 0.795 yes yes 5 2.76 0.141 0.58 0.591 yes yes
63
It can be noticed from Table 3.11, that the functions for the measured forces with
respect to the cutting speed are either ‘single’ or ‘parallel’ lines, with minimal
difference in the intercepts. This indicates that both the Fcm–t and Ftm-t functions are
only marginally affected by cutting speed V. This trend is the same as that for sharp
tool cutting and again confirms that cutting with tool flank wear does not affect the
trends of forces with respect to cutting speed.
Linear regression with covariance analysis has been done to study the effect of
wearland size on the slopes and intercepts of the regression lines for cutting forces
with respect to cut thickness. The results are given in Tables 3.12 and 3.13, from
which, it has been found that there is no significant difference (P≥0.05) between the
slopes for most of cases where tool flank wear is present at different wearland sizes,
i.e. the measured cutting force slope in the force-cut thickness diagrams is not
affected by tool flank wear. In contrast, the statistical results show that there is a
highly significant correlation between the intercepts in the force-cut thickness
diagram and the wearland size. Specifically, it has been found that the intercept
increases with an increase in the wearland size. The result has confirmed that tool
flank wear increased the rubbing force on the cutting edge/wearland. However, it is
still not clear if tool flank wear has affected the basic cutting process, i.e. the increase
in the measured cutting forces is merely due to the increase in the rubbing force, or a
combination of the increase in the rubbing force and the forces required generating
the shearing process. This will be studied in the next section.
Table 3.12 Covariance analysis of the effect of wearland size on the slopes of measured power force Fcm vs cut thickness t
Power force slopes intercepts Parallel Single
Cutting speed (mm/min) rake angle F P F P
100 -5 0.26 0.853 2.58 0.126 yes yes 0 2.63 0.132 12.38 0.003 yes no 5 25.94 0.004 12.44 0.017 no no
150 -5 0.61 0.626 3.49 0.07 yes yes 0 2.65 0.13 10.95 0.005 yes no 5 1.37 0.329 10.72 0.005 yes no
200 -5 0.77 0.541 9.82 0.005 yes no 0 1.03 0.434 6 0.024 yes no 5 3.87 0.075 9.42 0.011 yes no
64
Table 3.13 Covariance analysis of the effect of wearland size on the slopes of measured thrust force Ftm vs cut thickness t
Thrust force slopes intercepts Parallel Single
Cutting speed (mm/min) rake angle F P F P
100 -5 0.98 0.451 3.12 0.088 no yes 0 3.19 0.093 7.19 0.015 yes no 5 3.46 0.131 2.55 0.194 yes yes
150 -5 0.84 0.507 2.31 0.153 yes yes 0 3.88 0.056 8.41 0.007 yes no 5 4.02 0.07 10.15 0.009 yes no
200 -5 1.29 0.341 4.5 0.04 yes no 0 1.41 0.317 16.99 0.001 yes no 5 5.37 0.031 2.66 0.129 yes yes
3.4 The basic cutting quantities in the shear zone and tool chip interface
The basic cutting quantities, such as the shear angle or chip length ratio and shear
stress in the shear zone and friction angle in the tool-chip interface, are the
fundamental factors that character the basic cutting or chip formation process in
orthogonal cutting. The study of these basic cutting quantities enables the assessment
of the affect of tool flank wear on the basic chip formation process.
In machining or orthogonal cutting, it is not unreasonable to assume that when the
cut thickness is zero, the tool cuts nothing and, as such, the cutting forces should be
zero. However, no tool is perfectly sharp and there is rubbing between the tool and
workpiece surface, particularly when tool wear is present. Thus, even if the cut
thickness is zero, the cutting process still involves a cutting force (or rubbing force)
and this rubbing force increases with an increase in the wearland size. This has been
identified in the foregoing analysis in the previous section. It can be assumed that
this rubbing force is a result of the rubbing or ploughing process on the cutting edge
and wearland. According to Armarego [5], when evaluating the basic cutting
quantities (friction angle β, and shear stress τ), the rubbing force components must
be removed from the measured forces, so that only the forces on the shear zone are
used, namely,
65
Fcs = Fcm - Fcint (3.12)
Fts = Ftm - Ftint (3.13)
Where: Fcint and Ftint (N) are the forces corresponding to the intercepts in the
measured forces – cut thickness plots, i.e. the rubbing force component along the
power and thrust force directions, respectively.
In this section, the characteristics of the basic cutting quantities will be analysed and
compared with those for sharp tool and for different wearland sizes with an aim to
assess if tool flank wear affects the basic chip formation process for the reasons
given in Section 3.2. Based on the finding of this analysis, an approach to
establishing the predictive models for cutting performance will then be proposed in
the following sections.
3.4.1 The characteristics of chip length ratio 3.4.1.1 Effect of cut thickness and rake angle on the chip length ratio
Figures 3.8(a) and 3.8(b) show the typical trends on the effect of cut thickness on the
chip length ratio for both ‘sharp’ tools and tools with a wearland. From the figures, it
can be found that cut thickness does not significantly affect the chip length ratio.
This trend can be applied to other cases with different wearland sizes and is the same
as that found in previous studies of orthogonal cutting [5].
VB=0mm, V=150m/min
0
0.2
0.4
0.6
0.8
0 0.1 0.2 0.3 0.4
Cut thickness t (mm)
Chi
p le
ngth
ratio γ=−5
γ=0γ=5
VB=0.4mm, V=150m/min
0
0.2
0.4
0.6
0.8
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Chi
p le
ngth
ratio γ=−5
γ=0γ=5
(a) (b)
Figure 3.8 Effect of cut thickness on chip length ratio under different wearland sizes.
66
From Figure 3.8(a), it can be seen that the chip length ratio increases slightly with an
increase in rake angle when cutting with a sharp tool. It can also be noted that Figure
3.8(b) exhibits the same trend as the tool with a wearland of 0.4mm. A statistical
analysis of the experimental data has found that the presence of tool flank wear does
not change the way in which rake angle affects the chip length ratio (or shear angle).
Correlation tests of chip length ratio (rl) with cut thickness (t) at a given cutting
speed (V) and rake angle (γ) have also been carried out. The results are summarised
in Table 3.14.
Table 3.14 Correlation analysis of chip length ratio on cut thickness.
VB V γ Correlation coefficient
Test statistics for intercept
Test statistics for slope
Significance at 95% confidence level
t P-value t P-value Intercept
Dep. On ‘t’
Correl. Coef
0 100 5 0.497 10.842 0.008 0.810 0.503 Yes No No 0 0.666 12.757 0.006 1.264 0.334 Yes No No -5 0.982 20.268 0.002 7.363 0.018 Yes Yes Yes 150 5 0.769 18.595 0.003 1.701 0.231 Yes No No 0 0.490 26.873 0.001 0.795 0.510 Yes No No -5 0.865 23.049 0.002 2.437 0.135 Yes No No 200 5 -0.630 5.723 0.029 -1.148 0.370 Yes No No 0 0.914 31.618 0.001 3.181 0.086 Yes No Yes -5 0.590 17.076 0.003 1.035 0.410 Yes No No
0.2 100 5 -0.933 39.408 0.001 -3.656 0.067 Yes No Yes 0 -0.266 16.036 0.004 -0.39 0.734 Yes No No -5 0.943 6.038 0.026 4.007 0.057 Yes No Yes 150 5 0.854 26.229 0.001 2.323 0.146 Yes No No 0 -0.122 15.040 0.004 -0.174 0.878 Yes No No -5 0.998 106.715 0.000 20.15 0.002 Yes Yes Yes 200 5 0.909 19.048 0.003 3.085 0.091 Yes No No 0 0.538 7.147 0.019 0.903 0.462 Yes No No -5 0.961 30.863 0.001 4.899 0.039 Yes Yes Yes
0.4 100 5 0.648 14.750 0.005 1.204 0.352 Yes No No 0 0.869 22.331 0.028 1.755 0.330 Yes No No -5 0.935 33.731 0.001 3.722 0.065 Yes No No 150 5 0.425 25.328 0.002 0.664 0.575 Yes No No 0 0.848 32.081 0.001 2.259 0.152 Yes No No -5 0.694 17.327 0.003 1.363 0.306 Yes No No 200 5 0.848 13.207 0.006 2.263 0.152 Yes No No 0 0.413 16.434 0.004 0.641 0.587 Yes No No -5 0.630 15.06 0.004 1.147 0.370 Yes No No
0.6 100 5 -0.544 11.147 0.008 -0.918 0.456 Yes No No 0 -0.627 10.764 0.009 -1.138 0.373 Yes No No -5 -0.1 5.367 0.033 -0.142 0.9 Yes No No 150 5 -0.154 24.694 0.002 -0.220 0.846 Yes No No 0 0.518 18.633 0.003 0.858 0.482 Yes No No -5 0.645 5.642 0.30 1.194 0.355 No No No 200 5 0.969 35.773 0.001 5.527 0.031 Yes Yes Yes 0 0.319 10.320 0.009 0.476 0.681 Yes No No -5 0.887 17.771 0.003 2.713 0.113 Yes No No
67
From the table, it can be seen that at a 95% confidence level, the slopes are
significantly independent of the cut thickness for cutting with and without flank
wear. This analysis result confirms that chip length ratio is independent of the cut
thickness and tool flank wear does not change this trend.
An analysis of variance (ANOVA) has been conducted to quantitatively study the
effect of rake angle on chip length ratio. The statistical results are given in Table
3.15. It is shown that chip length ratio for the different rake angles could be
considered to be significantly different at 95% confidence level for cutting by tools
with and without flank wear. This finding is quantitative evidence that tool flank
wear does not affect this basic cutting quantity (i.e. shear angle) in orthogonal
cutting.
Table 3.15 ANOVA for the effect of rake angle on chip length ratio.
Wearland VB Speed V DOF Test Statistics Test at 95% Confidence Level (mm) (m/min) F-value P-value Equal ?
0 100 3,12 4.744 0.054 Yes 0 150 3,12 34.318 0.000 No 0 200 3,12 0.570 0.468 Yes
0.2 100 3,12 8.706 0.015 No 0.2 150 3,12 38.230 0.000 No 0.2 200 3,12 12.998 0.005 No 0.4 100 3,12 19.126 0.002 No 0.4 150 3,12 34.151 0.000 No 0.4 200 3,12 9.116 0.013 No 0.6 100 3,12 0.975 0.347 Yes 0.6 150 3,12 12.628 0.005 No 0.6 200 3,12 12.502 0.005 No
3.4.1.2 Effect of cutting speed on chip length ratio and shear angle
Figures 3.9(a) and 3.9(b) show the effect of cutting speed on chip length ratio when
cutting with both sharp tools and tools with flank wear. It can be noted that the effect
of cutting speed on chip length ratio is not significant, either for cutting using a sharp
tool or using a tool with flank wear. The trend between chip length ratio and cutting
speed found in the current study is consistent with previous investigations [5]. This
finding also indicates that the presence of tool flank wear does not affect the way in
which cutting speed affects the chip length ratio in orthogonal cutting.
68
0.0
0.2
0.4
0.6
0.8
1.0
50 100 150 200 250Cutting speed (m/min)
Chi
p le
ngth
ratio
γ=−5γ=0γ=5
VB=0mm, t=0.24mm
0.0
0.2
0.4
0.6
0.8
1.0
50 100 150 200 250Cutting speed (m/min)
Chi
p le
ngth
ratio
γ=−5γ=0γ=5
VB=0.2mm, t=0.17mm
(a) (b)
Figure 3.9 Effect of cutting speed on chip length ratio under different rake angles.
To further examine how tool flank wear affects chip formation process, the effect of
cutting speed on the shear angle is plotted in Figure 3.10. In Figure 3.10(a), it can be
found that shear angle slightly increases with an increase in cutting speed in exactly
the same manner as chip length ratio. A similar trend is also observed for cutting
with tool flank wear as shown in Figure 3.10(b).
0
10
20
30
40
50 100 150 200 250Cutting speed (m/min)
She
ar a
ngle
(deg
)
γ=−5γ=0γ=5
VB=0mm, t=0.24mm
0
10
20
30
40
50 100 150 200 250Cutting speed (m/min)
She
ar a
ngle
(deg
)
γ=−5γ=0γ=5
VB=0.2mm, t=0.17mm
(a) (b)
Figure 3.10 Shear angle vs cutting speed.
The results of ANOVA conducted to test the relationship between shear angle and
cutting speed are given in Table 3.16. It can be seen that for most of the cases
investigated in this study, the mean values are equal, which indicates that the effect
of cutting speed on shear angle is not significant and tool flank wear does not affect
this trend in orthogonal cutting.
69
Table 3.16 ANOVA results for the relationship between cutting speed and shear angle.
VB Rake angle F DOF P (95%) Equal
0 -5 3.130 2, 9 0.093 Yes 0.2 -5 1.471 2, 9 0.280 Yes 0.4 -5 3.486 2, 9 0.076 Yes 0.6 -5 0.778 2, 9 0.488 Yes 0 0 9.259 2, 9 0.007 No
0.2 0 7.040 2, 9 0.014 No 0.4 0 7.672 2, 9 0.017 No 0.6 0 1.717 2, 9 0.234 Yes 0 5 1.613 2, 9 0.252 Yes
0.2 5 6.954 2, 9 0.015 No 0.4 5 1.488 2, 9 0.277 Yes 0.6 5 1.474 2, 9 0.279 Yes
3.4.1.3 The effects of wearland size on chip length ratio and shear angle
Figures 3.11(a) and (b) show the relationship between chip length ratio and cut
thickness and between shear angle and cut thickness, respectively, under different
wearland sizes. From the figures, it can be noted that the trend lines for different
wearland sizes almost fall into a single line, which indicates that the effect of tool
wearland on both chip length ratio and shear angles is too small to be discernible
under this qualitative analysis.
0.0
0.2
0.4
0.6
0.8
1.0
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Chi
p le
ngth
ratio VB=0mm VB=0.2mm
VB=0.4mm VB=0.6mm
v=150m/min, γ=0°
0
10
20
30
40
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
She
ar a
ngle
(deg
)
VB=0mm VB=0.2mmVB=0.4mm VB=0.6mm
v=150m/min, γ=0°
(a) (b)
Figure 3.11 Effect of tool flank wear on chip length ratio and shear angle.
To quantitatively assess the effect of wearland on shear angle, an ANOVA was again
carried out to examine the mean values of shear angle for each cutting speed and rake
angle. This analysis confirmed the findings of the above qualitative analysis. The
results show that more than 89% of the cases tested possess equal mean values for
shear angle and flank wear and cut thickness; but these are not significantly
70
correlated at a 95% confidence level. Details of the ABOVA results are given in Table
3.17.
Table 3.17 ANOVA results for the effects of tool flank wear on shear angle.
Speed V Rake γ F DOF p(95%) Means equal?
100 -5 2.769 3, 12 0.087494 Yes 100 0 5.007 3, 11 0.019831 No 100 5 3.229 3, 12 0.060912 Yes 150 -5 0.263 3, 12 0.85042 Yes 150 0 1.298 3, 12 0.320175 Yes 150 5 0.314 3, 12 0.814811 Yes 200 -5 2.016 3, 12 0.165492 Yes 200 0 2.032 3, 11 0.167862 Yes 200 5 0.996 3, 12 0.427707 Yes
The fact that the wearland VB does not affect the shear angle (chip length ratio)
indicates that the effect of tool flank wear on the shear zone is not significant. This
feature further suggests that the shear process is almost independent of the wearland.
3.4.2 The characteristics of shear stress 3.4.2.1 Effect of cut thickness and rake angle on the shear stress
The effect of cut thickness on the shear stress in orthogonal cutting is shown in
Figures 3.12(a) and 3.12(b). It is noted that cut thickness has no significant effect on
shear stress in the cutting with ‘sharp’ tool (zero wearland), as shown in Figure
3.12(a). Similar trends have also been found in the cutting with a tool flank wear of
0.4mm, as shown in Figure 3.12(b). From the experimental data, this is a general
trend for all the tests. Thus, tool flank wear does not appear to affect the shear stress
in orthogonal cutting.
VB=0mm, V=200m/min
0
200
400
600
800
1000
0 0.1 0.2 0.3 0.4
Cut thickness t (mm)
She
ar s
tress
(N/m
m2 )
γ=−5γ=0γ=5
VB=0.4mm, V=150m/min
0
200
400
600
800
1000
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
She
ar s
tress
(N/m
m2 )
γ=−5γ=0γ=5
(a) (b)
Figure 3.12 Effect of cut thickness on shear stress under different rake angles.
71
The effect of rake angle on the shear stress can also be found from Figure 3.12. For
both sharp tools and tools with flank wear, the shear stress does not show any
discernible pattern with the change of rake angle, since the lines representing
different rake angles almost fall into a single line. It can thus be deduced that tool
flank wear does not affect the trend in which the cut thickness affects the shear
stress. To further confirm this finding, an ANOVA was carried out and the associated
results are given in Table 3.18, which shows that the mean values of shear stress are
equal for different wearland sizes.
Table 3.18 ANOVA for the effect of tool flank wear on the relationship between shear stress and rake angle.
Wearland VB Speed V DOF Test Statistics Test at 95% Confidence Level (mm) (m/min) F-value P-value Equal mean value?
0 100 3,12 4.663 0.056 Yes 0 150 3,12 1.756 0.215 Yes 0 200 3,12 0.652 0.438 Yes
0.2 100 3,12 6.561 0.028 No 0.2 150 3,12 4.371 0.063 Yes 0.2 200 3,12 1.055 0.329 Yes 0.4 100 3,12 0.404 0.541 Yes 0.4 150 3,12 7.391 0.022 No 0.4 200 3,12 0.427 0.528 Yes 0.6 100 3,12 2.468 0.147 Yes 0.6 150 3,12 13.053 0.005 No 0.6 200 3,12 2.114 0.177 Yes
A correlation analysis has been carried out to test the relationship between shear
stress and cut thickness in orthogonal cutting with and without tool flank wear. The
results are shown in Table 3.19, which indicate that the slope of regression lines is
significantly independent of the variation of cutting thickness at 95% confidence
level for all cases. The flank wear did not affect this relationship.
72
Table 3.19 Correlation analysis of shear stress on cut thickness.
VB V γ Correlation Coefficient
Test Statistics for intercept
Test Statistics for slope
Significance at 95% Confidence Level
t
P-value t
P-value Intercept slope Correl. Coef
0 100 5 0.493 15.885 0.004 0.801 0.507 Yes No No 0 0.855 44.975 0.000 2.331 0.145 Yes No No -5 0.887 12.018 0.007 2.710 0.113 Yes No No 150 5 0.766 21.382 0.002 1.686 0.234 Yes No No 0 0.136 24.443 0.002 0.194 0.864 Yes No No -5 0.849 25.475 0.002 2.270 0.151 Yes No No 200 5 -0.648 7.582 0.017 -1.203 0.352 Yes No No 0 0.921 49.952 0.000 3.344 0.079 Yes No No -5 0.658 77.036 0.000 1.235 0.342 Yes No No
0.2 100 5 -0.042 7.742 0.016 -0.60 0.958 Yes No No 0 -0.665 166.236 0.000 -1.261 0.335 Yes No No -5 0.892 6.141 0.026 2.792 0.108 Yes No No 150 5 0.771 24.247 0.002 1.710 0.279 Yes No No 0 -0.922 161.644 0.000 -3.360 0.078 Yes No No -5 0.824 7.235 0.019 2.055 0.176 Yes No No 200 5 0.853 21.911 0.002 2.311 0.147 Yes No No 0 0.487 13.826 0.005 0.788 0.513 Yes No No -5 0.924 27.296 0.001 3.420 0.076 Yes No No
0.4 100 5 0.631 10.481 0.009 1.149 0.369 Yes No No 0 0.088 10.475 0.061 0.089 0.944 No No No -5 0.511 7.316 0.018 0.840 0.489 Yes No No 150 5 0.695 93.343 0.000 1.367 0.305 Yes No No 0 0.849 36.332 0.001 2.270 0.151 Yes No No -5 0.816 31.278 0.001 2.510 0.124 Yes No No 200 5 0.930 41.622 0.001 3.584 0.07 Yes No No 0 0.355 4.161 0.053 0.537 0.645 No No No -5 0.754 22.503 0.002 1.625 0.246 Yes No No
0.6 100 5 -0.16 6.468 0.023 -0.229 0.840 Yes No No 0 -0.855 37.266 0.001 -2.333 0.145 Yes No No -5 -0.449 6.912 0.02 -0.71 0.551 Yes No No 150 5 -0.379 5.642 0.03 -0.579 0.621 Yes No No 0 0.390 9.833 0.01 0.598 0.610 Yes No No -5 0.694 9.116 0.012 1.363 0.306 Yes No No 200 5 0.343 1.330 0.315 0.517 0.651 No No No 0 0.122 25.776 0.002 0.174 0.878 Yes No No -5 0.742 12.388 0.006 1.565 0.258 Yes No No
3.4.2.2 Effect of cutting speed on shear stress
The effect of cutting speed on the shear stress from the orthogonal cutting
experiments is shown in Figures 3.13(a) and 3.13(b). From the figures, it is noted
that the effect of cutting speed on shear stress is marginal, although in some cases,
there is a slight increase of shear angle with an increase in cutting speed. This
relationship was further confirmed by a statistical analysis, as shown in Table 3.20.
From these analyses, the presence of tool flank wear does not affect the way in which
the cutting speed affects the shear stress both qualitatively and quantitatively. The
shear stress remains unchanged for cutting with both sharp tools and tools with flank
wear under corresponding cutting conditions.
73
0
200
400
600
800
50 100 150 200 250
Cutting speed (m/min)
She
ar s
tress
(N/m
m2 )
γ=−5γ=0γ=5
VB=0mm, t=0.17mm
0
200
400
600
800
50 100 150 200 250Cutting speed (m/min)
She
ar s
tress
(N/m
m2 )
γ=−5γ=0γ=5
VB=0.2mm, t=0.17mm
(a) (b)
Figure 3.13 Effect of cutting speed on shear stress under different rake angles.
Table 3.20 ANOVA for the effect of cutting speed on shear stress.
VB Rake angle F DOF p(95%) Equal 0 -5 15.288 2, 9 0.001276 No
0.2 -5 3.089 2, 9 0.095215 Yes 0.4 -5 4.377 2, 9 0.047014 no 0.6 -5 2.559 2, 9 0.131864 Yes 0 0 18.420 2, 9 0.000658 No
0.2 0 21.098 2, 9 0.0004 No 0.4 0 11.441 2, 7 0.006221 No 0.6 0 17.838 2, 9 0.000739 No 0 5 1.183 2, 9 0.349744 Yes
0.2 5 0.717 2, 9 0.513946 Yes 0.4 5 5.153 2, 9 0.03224 No 0.6 5 18.790 2, 9 0.000613 No
3.4.2.3 Effects of wearland size on shear stress
The trend of shear stress with respect to cut thickness is shown in Figures 3.14(a) and
3.14(b) under different wearland sizes. From the figures, no clear trends of shear
stress and wearland sizes can be observed, although there are a few cases where
shear stress varies with flank wear. The results of the corresponding ANOVA have
been carried out and the results are given in Table 3.21. There are about 50% of
cases that show no significant correlation between shear stress and wearland size at
95% confident level and about 70% of the cases that show no significant correlation
with wearland size at 99% confidence level. Consequently, it may be concluded that
tool flank wear has no effect on the shear stress.
74
0
200
400
600
800
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
She
ar s
tress
(N/m
m2 )
VB=0mm VB=0.2mm
VB=0.4mm VB=0.6mm
v=100m/s, γ=0°
0
200
400
600
800
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
She
ar s
tress
(N/m
m2 )
VB=0mm VB=0.2mm
VB=0.4mm VB=0.6mm
v=200m/s, γ=0°
(a) (b)
Figure 3.14 Effect of cut thickness on shear stress under different wearland sizes.
Table 3.21 ANVOA for the effect of tool flank wear on shear stress.
Cutting speed Rake angle F DOF p P(95%) Equal P(99%) Equal 100 -5 1.041795 3, 12 0.409335 Yes Yes 100 0 15.31289 3, 11 0.000305 No No 100 5 0.58964 3, 12 0.633434 Yes Yes 150 -5 5.593717 3, 12 0.012341 No Yes 150 0 29.66376 3, 12 7.82E-06 No No 150 5 29.13739 3, 12 8.59E-06 No No 200 -5 3.376655 3, 12 0.05444 Yes Yes 200 0 5.783483 3, 11 0.01266 No Yes 200 5 2.554881 3, 12 0.104279 yes Yes
3.4.3 The characteristics of friction angle 3.4.3.1 Effect of cut thickness and rake angle on the friction angle
The trends of friction angle β with respect to cut thickness are shown in Figures
3.15(a) and (b) for cutting without and with tool flank wear, respectively. From
Figure 3.15(a), it can be found that the friction angles are almost independent of the
cut thickness when sharp tools are used. A similar trend can also be observed from
Figure 3.15(b) where cutting tools with flank wear were used. From this qualitative
analysis, tool flank wear does not seem to affect the friction angle.
Figure 3.15 also shows the effect of rake angle on the friction angle. It can be noted
that friction angle appears to increase with an increase in the rake angle for cutting
with and without tool flank wear in a similar way to that noted in the literature
review. Qualitatively, tool flank wear does not appear to have affected the relation
between rake angle and friction angle.
75
VB=0mm, V=100m/min
0
10
20
30
40
0.00 0.10 0.20 0.30 0.40Cut thickness t (mm)
Fric
tion
angl
e (d
eg)
γ=−5γ=0γ=5
VB=0.6mm, V=150m/min
0
10
20
30
40
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Fric
tion
angl
e (d
eg)
γ=−5γ=0γ=5
(a) (b) Figure 3.15 Friction angle against cut thickness under different rake angles.
Table 3.22 Correlation analysis of friction angle against cut thickness t.
VB V γ Correlation coefficient
Test statistics for intercept
Test statistics for slope
Significance at 95% confidence level
t
P-value t
P-value Intercept
Dep. On ‘t’
Correl. Coef
0 100 5 -0.319 8.209 0.015 -0.476 0.681 Yes No No 0 -0.308 11.098 0.008 -0.457 0.692 Yes No No -5 0.335 74.796 0.000 0.503 0.665 Yes No No 150 5 -0.167 4.393 0.048 -0.239 0.833 Yes No No 0 0.321 39.166 0.001 0.480 0.679 Yes No No -5 0.133 25.113 0.002 0.19 0.867 Yes No No 200 5 0.336 8.370 0.014 0.504 0.664 Yes No No 0 -0.215 29.708 0.001 -0.312 0.785 Yes No No -5 0.195 29.555 0.001 0.282 0.805 Yes No No
0.2 100 5 0.261 170157 0.003 0.382 0.739 Yes No No 0 -0.212 16.356 0.004 -0.399 0.728 Yes No No -5 -0.911 26.772 0.001 -3.123 0.089 Yes No No 150 5 0.262 78.155 0.000 0.384 0.738 Yes No No 0 -0.030 37.025 0.001 -0.043 0.97 Yes No No -5 0.137 7.17 0.019 0.564 0.63 Yes No No 200 5 0.255 14.992 0.004 0.373 0.745 Yes No No 0 0.293 13.360 0.006 0.433 0.707 Yes No No -5 -0.967 109.402 0.000 -5.339 0.033 Yes No No
0.4 100 5 0.330 16.015 0.004 0.494 0.670 Yes No No 0 0.245 38.180 0.017 0.252 0.843 Yes No No -5 0.357 17.068 0.003 0.541 0.643 Yes No No 150 5 0.338 112.278 0.000 0.507 0.662 Yes No No 0 0.298 24.988 0.002 0.442 0.702 Yes No No -5 0.324 7.091 0.019 0.485 0.676 Yes No No 200 5 0.321 13.286 0.006 0.479 0.679 Yes No No 0 0.409 -0.459 0.691 0.633 0.591 N No No -5 0.356 2.956 0.098 0.539 0.644 N No No
0.6 100 5 0.405 14.633 0.005 0.627 0.595 Yes No No 0 -0.020 13.025 0.006 -0.028 0.980 Yes No No -5 -0.157 6.108 0.026 -0.225 0.843 Yes No No 150 5 -0.3 20.814 0.002 -0.445 0.7 Yes No No 0 -0.305 89.875 0.00 -0.453 0.695 Yes No No -5 0.311 11.145 0.008 0.462 0.689 Yes No No 200 5 -0.108 83.515 0.000 -0.153 0.892 Yes No No 0 -0.317 17.934 0.003 -0.473 0.683 Yes No No -5 0.325 30.145 0.001 0.485 0.675 Yes No No
76
To quantitatively study the effect of cut thickness and rake angle on the friction
angle, a statistical analysis has been conducted. The results are given in Tables 3.22
and 3.23. Table 3.22 shows that there is no correlation between friction angle β and
cut thickness t at 95% confidence level for all cases tested. The results in Table 3.23
show that the vast majority of the cases have unequal mean values and confirm the
fact that friction angle varies with the tool rake angle, while the presence of tool
flank wear does not affect the way in which the rake angle affects the friction angle.
Table 3.23 ANOVA for the effect of rake angle γγγγ on friction angle ββββ
Wearland VB Speed V DOF Test Statistics Test at 95% Confidence Level (mm) (m/min) F-value P-value Equal?
0 100 3,12 7.831 0.019 No 0 150 3,12 6.928 0.025 No 0 200 3,12 19.324 0.001 No
0.2 100 3,12 1.331 0.275 Yes 0.2 150 3,12 0.046 0.834 Yes 0.2 200 3,12 36.421 0.000 No 0.4 100 3,12 0.380 0.553 Yes 0.4 150 3,12 34.419 0.000 No 0.4 200 3,12 1.105 0.318 Yes 0.6 100 3,12 15.986 0.003 No 0.6 150 3,12 104.968 0.000 No 0.6 200 3,12 11.067 0.008 No
3.4.3.2 Effect of cutting speed on friction angle
The effect of cutting speed on friction angle is plotted in Figures 3.16(a) and 3.16(b).
0
20
40
60
50 100 150 200 250Cutting speed (m/s)
Fric
tion
angl
e (d
eg)
γ=−5γ=0γ=5
VB=0mm, t=0.31mm
0
20
40
60
50 100 150 200 250Cutting speed (m/s)
Fric
tion
angl
e (d
eg)
γ=−5γ=0γ=5
VB=0.6mm, t=0.31mm
(a) (b)
Figure 3.16 Friction angle vs cut speed under different rake angles.
From the figures, it can be noticed that when cutting with relatively low speed (100
to 150mm/min), the variation of friction angle with cutting speed is very marginal for
77
both sharp tools and the tools with flank wear. As the cutting speed further increases,
there is a slight decrease in friction angle.
The statistical analysis given in Table 3.24 shows that the mean values for different
cutting speeds are not equal, which confirms that cutting speed indeed has some
effect on the friction angle.
Table 3.24 Results of ANOVA for the effect of cutting speed on friction angle ββββ.
VB Rake F DOF P (95%) Equal 0 -5 259.556 2, 9 1.1E-08 No
0.2 -5 18.305 2, 9 0.000673 No 0.4 -5 32.267 2, 9 7.85E-05 No 0.6 -5 3.147 2, 9 0.092014 yes 0 0 75.523 2, 9 2.37E-06 No
0.2 0 16.709 2, 9 0.000933 No 0.4 0 245.572 2, 7 3.29E-07 No 0.6 0 25.637 2, 9 0.000192 No 0 5 5.810 2, 9 0.023976 No
0.2 5 28.158 2, 9 0.000134 No 0.4 5 10.281 2, 9 0.004741 No 0.6 5 50.04 2, 9 1.33E-05 No
3.4.3.3 The effects of wearland size on friction angle
Figures 3.17(a) and (b) show the effect of cut thickness on friction angle under
different wearland sizes. It can be seen that the effect of wearland size on the friction
angle is not discernible. This trend has been further confirmed through the ANOVA,
as given in Table 3.25. Consequently, tool flank wear does not affect the friction
angle in the tool-chip interface.
0
15
30
45
60
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Fric
tion
angl
e (d
eg)
VB=0mm VB=0.2mm
VB=0.4mm VB=0.6mm
v=150m/s, γ=0°
0
15
30
45
60
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Fric
tion
angl
e (d
eg)
VB=0mm VB=0.2mm
VB=0.4mm VB=0.6mm
v=200m/s, γ=5°
(a) (b)
Figure 3.17 Effect of wearland size on friction angle.
78
Table 3.25 Results of ANVOA for the effect of tool flank wear on friction angle ββββ.
speed Rake F DOF P (95%) Equal
100 -5 16.16452 3, 12 0.000163 No 100 0 21.28939 3, 11 6.93E-05 No 100 5 11.97689 3, 12 0.00064 No 150 -5 13.87571 3, 12 0.000331 No 150 0 16.74779 3, 12 0.000138 No 150 5 3.964795 3, 12 0.035458 No 200 -5 12.4166 3, 12 0.000546 No 200 0 23.62739 3, 11 4.26E-05 No 200 5 2.496447 3, 12 0.109479 No
3.4.4 Shear angle relationship and characteristics In this section, shear angle relationship in orthogonal cutting with wearland will be
investigated. As discussed in the literature, the shear angle relationship is an
important characteristic in orthogonal cutting. Various relationships can be found in
the literature. The general form of the relationship reported by Armarego [4, 44] is:
)(21 γβφ −+= cc (3.14)
Where: φ is the shear angle (degree), β is the friction angle (degree), γ is rake angle
(degree), and c1 and c2 are constants (intercept and slope of linear relationship) for a
given tool-work material combination.
0
10
20
30
15 25 35 45β−γ (degree)
She
ar a
ngle
φ (d
egre
e)
VB=0mmVB=0.2mmVB=0.4mmVB=0.6mm
Figure 3.18 Shear angle φφφφ against (ββββ-γγγγ) for different wearland sizes.
The experimental results obtained from the cutting tests appear to satisfy the shear
angle relationship for both ‘sharp’ tools and the tools with flank wear, as showing in
Figure 3.18. From the figure, the relationship appears to be linear which is consistent
79
with the previous studies of classic orthogonal cutting. The presence of tool flank
wear does not appear to affect the shear angle relationship in this study.
Based on the experimental data, the constants in Eq. (3.14) have been statistically
determined and the resulting equation is given by:
φ = 25.597 – 0.263(β - γ) (3.15)
The statistical analysis showed that coefficient of determination (R2) for the above
equation is about 0.50 which confirms the linear relationship although many other
factors such as the cutting speed may have affected the linearity (note that this
regressions analysis involved a 30 degrees of freedom problem so that R2=0.50 may
be considered as significant to confirm the linearity). An attempt has also been made
to find the constants (intercept and slope) for the above equation for each wearland
size (including zero). It is interesting to note that the intercepts for all the wearland
sizes from the regression analysis are almost the same, so are the slopes, as shown in
Table 3.26. This again confirms that tool flank wear does not affect the shear angle
relation.
Table 3.26 Coefficients in shear angle relation under different wearland sizes.
VB (mm) 0 0.2 0.4 0.6 overall
Intercept 25.777 25.661 25.620 25.969 25.597
Slope -0.267 -0.278 -0.269 -0.263 -0.263
3.4.5 Summary
The foregoing analysis has shown that tool flank wear results in a significant increase
in the total cutting force. More importantly, this analysis has proved for the first time
that the flank wear of a cutting tool does not affect the basic cutting quantities both
qualitatively and quantitatively, and hence does not affect the forces required for chip
formation in the shear zone and tool-chip interface. It has confirmed that tool flank
wear results in an additional ploughing force on the wearland and thus an increase in
the overall cutting forces. The practical implications of this finding is significant in
80
that the thin shear zone analysis in early studies can still be used to determine the
forces in the shear plane and at the tool–chip interface for cutting with tool flank
while the additional rubbing or ploughing forces on the wearland are best considered
separately.
It is possible for the forces on the wearland to be modelled and determined by an
analytical approach, such as using the slip-line theory. Due to the limitation of the
work in this thesis, the wearland forces will be experimentally determined as a
function of the orthogonal cutting variables for a given tool-work material
combination. For this purpose, ploughing or rubbing forces, i.e. that forces when the
cut thickness is zero, will be analyzed in the next section.
3.5 Analysis of the rubbing force components
Based on the detailed investigation presented in the previous section, tool flank wear
results in a rubbing force component, which contributes to the increase in the overall
cutting forces in orthogonal cutting. It is now necessary to study the rubbing force
and understand where the force is generated and how it is related to the cutting or
process parameters. From the foregoing analysis, it has become apparent that a
rubbing force exists as evidenced by the force noticed at zero cut thickness (i.e. the
intercept in the force versus cut thickness plots), and the rubbing force increases with
an increase in the wearland size. It has been reported in early investigations that even
for ‘sharp’ tools, the rubbing or ploughing force still exists and such force has been
defined as the ‘edge force’ component [5, 39]. Thus, the overall cutting forces for
‘sharp’ tool cutting arise from two sources, namely from the work deformation and
the edge effect [5]. The edge force has been commonly determined from experiments
and expressed as a function of edge force intensity and cut width, i.e.
bcF ee ⋅= (3.16)
Where: the ce (N/mm) is the edge force intensity found from cutting experiments.
When cutting with tool flank wear, the rubbing force is a combination of the ‘edge
force’ and the ploughing action between the wearland and work surface. Thus, it is
81
necessary to differentiate or separate the ‘edge force’ from the total rubbing force, so
the rubbing force on the wearland can be studied individually.
V=200m/min, γ=5 degree
0
200
400
600
800
0 0.2 0.4 0.6 0.8VB(mm)
Rub
bing
forc
e (N
)
FtintFcint
Figure 3.19 Rubbing force components vs wearland size.
Figure 3.19 shows the typical relationship between the two components of the total
rubbing force and the wearland size, where Fcint is the rubbing force component in
the cutting direction and Ftint is that in the thrust direction normal to the cutting
direction. It is apparent from the figure that the rubbing forces increase with an
increase in the wearland size. It should be noted that when VB=0, the rubbing force
is not zero, confirming the existence of the so-called ‘edge force’. Thus, removing
the edge force at zero wearland size will give the rubbing force corresponding to the
wearland size under consideration. These rubbing forces on the wearland, or the
wearland forces, can then be analysed.
In the previous section, it has been proven both qualitatively and by statistical
analysis that the measured cutting forces have a linear relationship with the cut
thickness. In other words, the change of cut thickness does not change the slope of
the linear relationship between the total cutting forces and cut thickness. Under
different wearland sizes, the plots of force and cut thickness appear to be parallel
lines. Consequently, the force intercept at zero cut thickness where the rubbing or
ploughing force is obtained is not affected by the cut thickness.
It is reasonable to assume that the magnitude of the wearland force is directly
proportional to the length of the cutting edge (or the width of cut, b) engaged in
82
cutting for a given wearland size, so that it may be considered together with wearland
size in the form of a ‘contact area’ (if assuming that the contact area between the tool
flank and the work surface is equal to the wearland area). Other variables, such as the
cutting speed and rake angle, will need to be considered in the analysis of wearland
force. Thus, the wearland force may be expressed as:
),V),VBb((fFw γ⋅= (3.17)
Where: Fw denote the wearland force (N), which includes two mutually perpendicular
components, Fcw and Ftw. From Figure 3.19, the rubbing force or wearland force and
the contact area are likely to be in linear relationship.
Before determining the relationship between the wearland force components and the
various cutting parameters, a statistical analysis is carried out to identify the variables
that significantly affect the rubbing force or wearland force. This analysis is shown
in Table 3.27.
Table 3.27 Correlation analysis of wearland force with process variables.
Input variables Wearland area (b VB)
Cutting speed V
Rake Angle γ
Correlation coefficient 0.872(*) 0.120 -0.127
Sig. (2-tailed) 0.000 0.544 0.521 Fcw Number of samples N 28 28 28
Correlation coefficient 0.728(*) 0.270 -0.421
Sig. (2-tailed) 0.000 0.164 0.026 Ftw Number of samples N 28 28 28
* At the 99% confidence level; all others are at 95% confidence level.
The correlation coefficient between Fcw and (b VB) is 0.872, and between Ftw and VB
is 0.728, while the corresponding significant values are zero. Therefore, there is a
strong correlation between the wearland force and contact area at the wearland. The
correlation analysis also shows the relationship between the wearland force and the
cutting speed as shown in Table 3.27. The correlation coefficients between the
wearland force and cutting speed are 0.12 for Fcw and 0.27 for Ftw, respectively, and
the significant values are all greater than 0.05. These results indicate that no
83
significant relationship exists between the wearland forces and cutting speed V.
Negative coefficients have been found between the wearland forces and rake angle,
which suggests negligible effects of rake angle on the wearland forces.
Consequently, the wearland force may be expressed as a function of the contact area
on the wearland (note that the contact area is assumed to be equal to the wearland
area). Thus, equation (3.17) can be re-written as:
)( VBbCF cwcw = (3.18)
)( VBbCF twtw = (3.19)
Where: Ccw and Ctw are wearland force coefficients (intensities) (N/mm2).
Based on the above analysis and the likely relationship between the wearland force
and wearland area or contact area, a regression analysis has been carried out to
determine the mathematical equations for the wearland force and edge force as well
at a 95% confidence level. While several other forms of models, such as second
order and exponential models exist, the linear model appears to give the identical
coefficient of determination. Thus, the linear model has been chosen.
For the purpose of regression analysis, the edge force and wearland force
components have been combined into linear equations as:
VBbCbCF cwcecrub += (3.20)
VBbCbCF twtetrub += (3.21)
The first term is the edge force component and the second term is the wearland force
in the equations. The constants in the equations have been statistically determined as
given in Table 3.28, where the coefficients of determination (R2) are also given. For a
case involving 28 observations, the linear regression criterion for the R2 value is
0.374. It can be seen from Table 3.28 that the R2 values for the two equations are
significantly greater than the criterion value. Thus the equations are adequately
describing the rubbing forces.
Table 3.28 Regression results for rubbing forces with VB, V and rake angle by SPSS
Regression model R2 Predictors: Coefficients Cce 73 VBbCbCF cwcecrub += 0.831 Ccw 179.8 Cte 86.1 VBbCbCF twtetrub += 0.575 Ctw 156.1
84
Thus, equations (3.20) and (3.21) can be re-written as:
VBbbFcrub 8.17973 += (3.22)
VBbbFtrub 1.1561.86 += (3.23)
When the wearland size is zero, the above equations give the edge forces only. These
empirical equations for the edge and wearland forces will be used in the predictive
models for the overall cutting force to be developed in the next section.
3.6 Proposed orthogonal cutting force models with tool flank wear
Predictive cutting force models will be developed for orthogonal cutting allowing for
the effects of flank wear. For this purpose, the database for the basic cutting
properties that are needed in the predictive models will also be established. The
models are then assessed for their predictive capability.
3.6.1 Model formulation Based on the analysis in previous sections, tool flank wear does not qualitatively and
quantitatively affect the basic chip formation process in orthogonal cutting, as
represented by the basic cutting quantities (shear angle, friction angle and shear
stress). Thus, the forces involved in the shearing process in the shear zone and in the
tool-chip interface can be predicted using an established machining theory in the
literature, such as the thin shear zone model. The analysis also has found that there is
a rubbing or ploughing force on the tool edge and tool wearland as evidenced by the
existing forces when the cut thickness is zero. This rubbing force increases with an
increase in the wearland size. When a ‘sharp’ tool is used, this rubbing force is in
fact the so-called ‘edge force’, while the additional amount resulting from the
wearland has been classified as wearland force. For the purpose of force prediction in
this thesis, empirical equations for the edge force and wearland force have been
established which combined with the forces required for the chip formation process,
can form the model to predict the total forces in orthogonal cutting with tool flank
wear.
85
Based on the above analysis, the overall force relationships and the model to predict
the forces are shown in Figure 3.20. Mathematically, the overall cutting force model
can be given by:
VBbCbCbtFFFF cwcecwcecsc ⋅++−+
−=++=)cos(sin
)cos(γβφφ
γβτ (3.26)
VBbCbCbtFFFF twtetwtetst ⋅++−+
−=++=)cos(sin
)sin(γβφφ
γβτ (3.27)
Fcs
Fts
FS
FN
θ
N
Rs’
F
Rs
β
φ
Re’
Rw’
Re
Fce
Fte
Rw
Fcw
Ftw
Plastic zones
Cutting Tool
Flank wear land zone
Chip
tWork
tc
γ
Rs: Shear forceRe: Edge forceRw: Flank wear force
R =Rs+Re+Rw
Fc=Fcs+Fce+Fcw
Ft=Fts+Fte+Ftw
Figure 3.20 Orthogonal cutting model allowing for the effect of tool flank wear.
It is noteworthy that the proposed cutting force models make full use of the
previously developed machining theories for ‘sharp’ tools. It is anticipated that the
proposed orthogonal cutting model can be mathematically related to oblique cutting
and various practical machining operations, such as turning operations, to develop
predictive force models. In order to implement the proposed force model, the basic
cutting quantities, such as the shear angle and friction angle, need to be determined.
86
3.6.2 Basic cutting quantities database The equations for the basic cutting quantities with respect to cutting process
variables are fitted from the experimental data. As shown previously in this chapter,
tool wearland VB does not affect the basic cutting quantities. A similar analysis has
been carried out and it found that the cut thickness does not affect the basic cutting
quantities either at a 95% confident level for both the Pearson Correlation and
Nonparametric Correlation, as shown in Table 3.29, where the analysis of the two
correlations for tool wearland are also given.
Table 3.29 Correlations of the basic cutting quantities with cutting variables.
Dependent Variable Correlation
method Dependent Variable
Analysis variables Cutting
speed V Rake
Angle γ VB Cut
thickness t Pearson
Correlation -.667(**) .511(**) -.147 -.065
Sig. (2-tailed) .000 .000 .111 .482 Friction angle β
N 119 119 119 119 Pearson
Correlation .180 .250(**) .133 .077
Sig. (2-tailed) .050 .006 .150 .406 shear stress
τ N 119 119 119 119
Pearson Correlation .465(**) .422(**) .255(**) .207
Sig. (2-tailed) .000 .000 .005 .024
Pearson Correlation
Chip length ratio rl
N 119 119 119 119 Correlation Coefficient -.678(**) .496(**) -.140 -.044
Sig. (2-tailed) .000 .000 .130 .635
Friction angle β
N 119 119 119 119
Correlation Coefficient .314(**) .258(**) .038 .094
Sig. (2-tailed) .001 .005 .682 .310 shear stress
τ N 119 119 119 119
Correlation Coefficient .467(**) .420(**) .236(**) .155
Sig. (2-tailed) .000 .000 .010 .091
Nonparametric Correlation
Chip length ratio rl
N 119 119 119 119 ** Correlation is significant at the 0.01 level (2-tailed).
The variations of each basic cutting quantity with respect to rake angle and cutting
speed have been considered and a series of regression equations for each cutting
87
quantity, i.e. rl, β and τ, have been established using multi-regression analysis. The
first order terms of the independent operation variables, namely, rake angle and
cutting speed, have finally been considered, because this form gives simpler
equations while their predictive capabilities are identical to other complex models.
The summary of the regression models and the associated statistical results are given
in Table 3.30, while Table 3.31 gives the final fitted equations together with the
coefficients for the edge and wearland forces established in the previous section.
Table 3.32 shows that the regression equations are adequate to predict these cutting
quantities.
With these basic cutting quantities and the edge force and wearland force
coefficients, it is possible to evaluate the cutting forces in orthogonal cutting by
using the proposed models.
Table 3.30 Regression Model Summary and ANOVA.
Model (Dependent Variable)
R Square Sum of
Squares df Mean Square F Sig.
Regression 1150.354 2 575.177 107.983 .000(a) Residual 617.879 116 5.327 Friction angle β 0.651
Total 1768.233 118 Regression 55833.999 2 27916.999 6.712 .002(a) Residual 482484.426 116 4159.348 Shear stress τ 0.104
Total 538318.424 118 Regression 0.030 2 .015 44.288 .000(a) Residual 0.039 116 .000 Chip length ratio rl 0.475
Total 0.070 118 Predictors: (Constant), Cutting speed V, Rake Angle γ and rl
Table 3.31 Database for basic cutting quantities.
Cutting quantity Equation Unit
Friction angle β β=36.51+0.43γ-0.06V Degree
Shear stress τ τ=413.52+4.43γ+0.34V N/mm2
Chip length ratio rl rl=0.287+0.003γ None
Cce=73 N/mm Edge force coefficients
Cce and Cte Cte=86.1 N/mm
Ccw=179.8 N/mm2 Wearland force coefficients
Ccw and Ctw Ctw=156.1 N/mm2
γ in degrees, V in m/min
88
Table 3.32 Coefficients of the regression lines.
Unstandardized Coefficients
Standardized Coefficients Model (Dependent
Variable) Name of
Coefficients Value Std. Error Beta
T Sig.
(Constant) 36.51 .817 44.671 .000 Rake angle γ .43 .052 .455 8.252 .000 Friction angle β
(degrees) Cutting speed V -.06 .005 -.626 -
11.367 .000
(Constant) 413.52 22.84 18.104 .000 Rake angle γ 4.43 1.46 .268 3.040 .003 Shear stress τ
(N/mm2) Cutting speed V .34 .148 .204 2.311 .023
(Constant) .287 .007 43.885 .000 Rake angle γ .003 .000 .467 6.659 .000 Chip length ratio
rl Cutting speed V .000 .000 .507 7.224 .000
3.6.3 Assessment of the predictive model The proposed force predictive models for orthogonal cutting with tool flank wear
will need to be verified for their plausibility and predictive capability. The model
verification is conducted by comparing both qualitatively and quantitatively the
predicted forces with the corresponding experimental results.
Figures 3.21(a) and 3.21(b) show the general trends of the predicted cutting forces
with respect to the cut thickness by solid lines. For comparison purposes, the
experimental data are also given in symbols in the figure. The predicted forces were
evaluated using equations 3.26 and 3.27, and the basic cutting quantity database in
Table 3.31. It can be seen that the forces increase linearly with the cut thickness and
are in good correlation with the experimental data. It has also been found that cutting
forces increase with the wearland size. Plots of cutting forces vs other variables
under different conditions have also been carried out and show that the predicted
trends of the cutting forces are consistent with those reported in the literature and
from the mechanics of cutting analysis. Thus, the generality and plausibility of the
proposed cutting force models have been amply demonstrated.
89
0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Pow
er fo
rce
Fc (
N) VB=0.6mm
VB=0.4mmVB=0.2mmVB=0mm
v=150m/min, γ=0°0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4Cut thickness t (mm)
Thru
st fo
rce
Ft (
N) VB=0.6mm
VB=0.4mmVB=0.2mmVB=0mm
v=150m/min, γ=0°
(a) (b)
Figure 3.21 Predicted and experimental cutting forces.
(a) power force Fc and (b) thrust force Ft.
Quantitative comparisons between the predicted and experimental forces have been
carried out to examine the adequacy of the models. The comparisons are based on the
percentage deviations of the models predicted values with respect to the
corresponding experimental data using the following equation:
100Pr
.% ×−=forcealExperiment
forcealExperimentforceedicteddev (3.31)
The histograms for the percentage deviation of the predicted cutting forces with
respect to the experimental data are given in Figures 3.22 and 3.23. It can be seen
from Figure 3.22 that the average and standard deviations for different wearland size
do not exhibit any differences that can suggest the effect of wearland on the model’s
predictive capability, and the models give good prediction for both the power and
thrust force components with the maximum average deviation being 2.6% for power
force and 4.7% for thrust force.
The histograms for overall comparison for all the wearland sizes including ‘sharp’
tools are given in Figure 3.23. It shows that the model’s prediction gives an average
percentage derivation of 0.5% with a standard deviation of 9.44% for the power
force, while the average deviations for the thrust force is -0.5% with a standard
deviation of 14.46%. Thus, it may be concluded that the predictive model can give
adequate prediction for the cutting forces in orthogonal cutting when tool flank wear
is present.
90
Mean= -2.5%Std. Dev=6.93%
Percentage deviation (Fc) VB=0mm
Freq
uenc
y %
Mean= -4.7%Std. Dev=13.66%
Percentage deviation (Ft) VB=0mm
Freq
uenc
y %
(a) (b)
Mean=2.5%Std. Dev=5.29%
Percentage deviation (Fc) VB=0.2mm
Freq
uenc
y %
Mean=0.4%Std. Dev=9.01%
Percentage deviation (Ft) VB=0.2mm
Freq
uenc
y %
(c) (d)
Mean= -0.7%Std. Dev=11.35%
Percentage deviation (Fc) VB=0.4mm
Freq
uenc
y %
Mean= -1.1%Std. Dev=18.73%
Percentage deviation (Ft) VB=0.4mm
Freq
uenc
y %
(d) (f)
91
Mean=2.6%Std. Dev=11.93%
Percentage deviation (Fc) VB=0.6mm
Freq
uenc
y %
Mean=3.3%Std. Dev=14.47%
Percentage deviation (Ft) VB=0.6mm
Freq
uenc
y %
(g) (h)
Figure 3.22 Histograms for percentage deviations between predicted and experimental force components under different wearland sizes.
Mean=0.5%Std. Dev.=9.44%
Precentage Deviation (Fc)
Freq
uenc
y %
Mean= -0.5 %Std. Dev.=14.46%
Precentage Deviation (Ft)
Freq
uenc
y %
(a) (b)
Figure 3.23 Histograms for percentage deviations between predicted and experimental force components for all wearland sizes.
3.7 Concluding remarks A comprehensive experimental investigation has been conducted into the orthogonal
cutting process allowing for the effect of tool flank wear. The investigation
considered a wide range of cutting parameters including cut thickness, cutting speed,
tool rake angle and wearland size when cutting mild carbon steel CS1020 with
carbide tools. The selection of wearland sizes was based on the ISO3685
International Standard on tool life and some were higher than the recommended
value to study the fore pattern for tool condition monitoring in future investigations.
92
The selection of other cutting variables was based on a common range of
applications and the machine tool limitations
The measured force components and chip length ratio as well as other derived cutting
properties, such as the friction angle and shear stress, have been qualitatively and
statistically analysed to study the effect of tool flank wear on the basic cutting or chip
formation process in orthogonal cutting. The qualitative trends and the detailed
statistical analysis have been analysed for the various basic cutting quantities with
respect to the process conditions tested and in comparison with their characteristics
in orthogonal cutting with ‘sharp’ tools. Qualitatively, the characteristics of the basic
cutting quantities (chip length ratio or φ, τ, β, and shear angle relation) possess
similar trends to those in orthogonal cutting with sharp tools. The outcome of the
study is further confirmed by a more comprehensive statistical analysis.
A major finding of the experimental analysis is that the tool flank wear does not
qualitatively or quantitatively affect any of the basic cutting quantities. There is
strong evidence that the tool flank wear does not affect the basic chip formation
process and, as such, the forces required to perform the chip formation in the shear
zone and tool-chip interface should be the same for cutting with sharp tools and with
tool flank wear. Thus, the increase in the overall cutting force is considered as a
result of the increased rubbing or ploughing force on the cutting edge and wearland.
The rubbing force was thoroughly investigated and it was found that it increases with
an increase in the wearland size. The rubbing force was then separated into an ‘edge
force’ component, and a ‘wearland force’ component. The characteristics of the
wearland force and its relationship with other process parameters were then analysed,
and the associated empirical equations for the wearland forces have been developed,
along with those for the edge forces. Before a more comprehensive fundamental
investigation is carried out to mode the edge and wearland forces, these empirical
equations are essential in developing the cutting force models for machining with
tool flank wear.
Based on the findings in this investigation, a predictive model for the cutting forces
in orthogonal cutting with tool flank wear has been proposed. This model together
93
with the introduction of edge and wearland forces makes full use of existing
machining theories to evaluate the forces required for the shearing process. In order
to implement the predictive model, a database of basic cutting quantities has been
established for the tool-work material combination considered in this study. The
plausibility and generality of the predictive model have been amply demonstrated,
while the quantitative comparison of the model predictions with the corresponding
experimental data showed that the average percentage deviation for the power force
and thrust force were 0.5% and -0.5% respectively, while the respective standard
deviations were 9.44% and 14.46%. Thus, the model can adequately predict the
cutting forces in orthogonal cutting with and without tool flank wear.
The proposed orthogonal cutting model will need to be extended to oblique cutting
and turning operations with tool flank wear in the following chapters.
94
Chapter 4 Predictive cutting force models for oblique cutting allowing for tool flank wear effects
Oblique cutting is a more general cutting process than orthogonal cutting as
described in the literature review. Similar to orthogonal cutting, oblique cutting
involves wedge shaped tool with a straight cutting edge to cut a workpiece of
constant width at a uniform cut thickness and constant resultant velocity. However,
the resultant cutting velocity is inclined at an acute angle to the plane normal to the
cutting edge. This acute angle is the inclination angle that distinguishes oblique
cutting from orthogonal cutting. The inclination angle plays a major role in
controlling the chip flow direction and resultant force in the oblique cutting process.
It results in a more complicated cutting process than orthogonal cutting, as reviewed
in Chapter 2. The orthogonal cutting may be considered as a special case of oblique
cutting where the inclination angle is zero. It is known that the oblique cutting
process is more closely related to many practical machining operations, such as
turning or milling; so that the development of an oblique cutting model is an
essential step towards modelling the cutting processes in practical machining
operations.
In this chapter, the study of orthogonal cutting with tool flank wear described in the
previous chapter will be extended to oblique cutting where a 3-D chip formation
process is involved. For this purpose, attention will be paid to the development of
cutting force models for the oblique cutting process and consideration given to the
effects of tool flank wear. The forces required to shear the work material and the
forces on the cutting edge (or edge force) and wearland (or wearland force) will be
analysed and applied to the 3-D cutting process before arriving at the final predictive
model. The models will then be experimentally verified. The plausibility and
adequacy of the models will be analysed by comparing the model prediction with the
corresponding experimental data.
95
4.1 Formation of the predictive cutting force model The analysis in this thesis is based on the Merchant type thin shear zone model which
has been reviewed in the Chapter 2. In addition, the commonly recognised rubbing
force on the cutting edge (or edge force) as well as the rubbing force on the wearland
as studied for orthogonal cutting will need to be taken into account in modelling the
oblique cutting process. For the purpose of modelling the cutting forces, the
assumptions made are the same as those in the orthogonal cutting discussed in the
previous chapter. In addition, the collinear conditions apply to the oblique cutting
process, i.e. the direction of friction force F on the rake face is collinear to the chip
velocity, the direction of shear force in the shear plane is collinear to the shear
velocity, and the rubbing force on the wearland is collinear with the cutting velocity.
As analysed in the previous chapter, the tool wearland has been confirmed to have no
effect on the basic cutting qualities in the orthogonal cutting, i.e. it does not affect the
forces required for chip formation. The edge and wearland forces have been
successfully separated from orthogonal cutting as additional force components, so
that the total force can be expressed as
wes FFFF ++= (4.1)
As oblique cutting involves a 3-D deformation process, the forces involved need to
be represented by three mutually perpendicular force components. Thus, a mechanics
of cutting analysis is yet to be made to model the force components in three mutually
perpendicular directions. All the forces in the oblique cutting are shown in the Figure
4.1. There are forces for basic chip formation (Fcs, Fts and Frs), edge forces (Fce, Fte
and Fre) and wearland forces (Fcw, Ftw and Frw). Before arriving at the models for the
overall cutting forces, the force from each source (or purpose) will be analysed
separately in the following sections.
96
Figure 4.1 Forces in oblique cutting
4.1.1 Forces required for basic chip formation
The models for the forces required to form the chip in the shear zone and tool-chip
interface are taken from the well known thin shear zone analysis [4, 5, 112]. The
three mutually perpendicular force components in the shear zone are denoted as the
power force Fcs, thrust force Fts, and radial force Frs, which are directly proportional
to the width of cut b and cut thickness t as reviewed in the literature. These force
components can be expressed by the linear equations below.
btKF cscs = (4.2)
btKF tsts = (4.3)
btKF rsrs = (4.4)
Where: the coefficients Kcs, Kts and Krs (N/mm2) are related with tool and workpiece
material properties and cutting conditions. They can be given by [1]
ncnnn
ncnn
ncs
iK
βηγβφβηγβ
φτ
222 sintan)(cos
sintantan)cos(sin ⋅+−+
⋅⋅+−⋅= (4.5)
97
ncnnn
nn
nts i
Kβηγβφ
γβφτ
222 sintan)(cos
)sin(cossin ⋅+−+
−⋅= (4.6)
ncnnn
ncnn
nrs
iK
βηγβφβηγβ
φτ
222 sintan)(cos
sintantan)cos(sin ⋅+−+
⋅−⋅−⋅= (4.7)
In addition, the variables in the above equations have the following relationships:
ncl
ncln ir
irγη
γηφsin)cos/(cos1
cos)cos/(costan
⋅⋅−⋅= (4.8)
cn ηββ costantan ⋅= (4.9)
ii
nc
nnn tansintan
costan)tan(
⋅−⋅=+
γηγβφ (4.10)
In implementing the above force equations, the chip length ratio rl, shear angle β and
shear stress τ are found from the database developed in orthogonal cutting. In doing
so, the normal rake angle γn is used as the rake angle in orthogonal cutting. The chip
flow angle c is assumed to be equal to the cutting edge inclination angle, i.e. c=i. In
using the models, it has been recognised by many researchers, such as Armarego [5]
that the chip flow angle in oblique cutting has some deviation from the inclination
angle, particularly for a larger inclination angle. To overcome this discrepancy, an
iteration searching procedure as proposed by Armarego [5, 44] is used, whereby the
inclination angle is used as a starting value for chip flow angle and the normal shear
angle and friction angle are calculated from the chip flow angle. The collinear
conditions between the friction force and chip flow direction as well as the shear
force and shear velocity from the obtained normal shear angle and normal friction
angle are then checked. This is done by checking if equation (4.10) exists. If the
collinear conditions are not satisfied, the chip flow angle is adjusted by a small
increment and the searching procedure starts again until the collinear conditions are
met. This numerical solution procedure provides the ‘predicted’ value of the chip
flow angle, normal shear angle and friction angle which increase the accuracy of the
predicted forces in oblique cutting.
98
4.1.2 Edge force component
The edge forces are illustrated in the Figure 4.2. It can be seen the total length of the
tool edge engaged in cutting is longer than that in orthogonal cutting for the same
width of cut b. The tool edge length engaged in oblique cutting is b/cos(i). The
resultant edge force Re’ is proposed to lie in the Pn plane. The edge force component
Fce’ is normal to the tool edge and the Fct’ is normal to the Vw in Figure 4.2.
Figure 4.2 Edge forces on the cutting edge in oblique cutting
The edge force is the function of the cutting edge length from the orthogonal cutting
analysis in the previous chapter. So the edge forces in oblique cutting can be
expressed as:
)cos(/' ibCF cece = (4.11)
)cos(/' ibCF tete = (4.12)
If the 'ceF is projected to the directions along and normal to the cutting speed V, two
new edge force components Fce and Fre are formed, and with the thrust force teF will
result in three mutually perpendicular force components as given below:
)cos(' iFF cece = (4.13)
'tete FF = (4.14)
)sin(' iFF cere = (4.15)
99
Substituting equations (4.11) and (4.12) gives:
bCF cece = (4.16)
)cos(/ ibCF tete = (4.17)
)tan(ibCF cere = (4.18)
Where: Cce and Cte (N/mm) are found from the basic cutting quantity database, and
width of cut b and inclination angle i are given variables.
4.1.3 Wearland force component It has been established in orthogonal cutting that the wearland force is proportional to
the contact area between the wearland and the work surface and the contact area is
approximated to be the area of wearland. In oblique cutting, the cutting edge is not
perpendicular to the cutting velocity; but is inclined at an acute angle to the normal to
the cutting edge. The two components of the rubbing force on the wearland or
wearland force are therefore in the cutting direction and normal to the cutting
direction. In addition, the width of cut is not normally equal to the engaged length of
the cutting edge. According to the International Standard (ISO 3685:1993), the width
of the wearland VB is measured in the tool cutting edge plane and perpendicular to
the cutting edge, so that the effective wearland size is in fact larger than the nominal
wearland size VB. The geometrical relationships are shown in Figure 4.3.
VB
b
i
i
VB
’
b’
V
Wearland
Workpiece
Figure 4.3 The contact area of weraland and workpiece in the oblique cutting.
100
From Figure 4.3, the effective contact length of cutting edge b’, and effective
wearland size VB’ can be given by:
)cos(/' ibb = (4.19)
)cos(/' iVBVB = (4.20)
If the collinearity condition applies, the wearland force in the cutting direction Fcw is
collinear to the cutting velocity, while the thrust component Ftw is normal to the
cutting velocity, as shown in Figure 4.4. The wearland force will not result in a third
force component.
)cos()(
0
'
iVBbC
dbVBCF cwb
bcwcw
=
== (4.21)
)cos()(
0
'
iVBbC
dbVBCF twb
btwtw
=
== (4.22)
0=rwF (4.23)
Where: Ccw and Ctw (N/mm2) are the wearland force intensity factors and can be
found from the basic cutting quantity database.
Figure 4.4 Wearland forces in oblique cutting
101
4.1.4 Overall cutting force models for oblique cutting with tool flank wear
According to equation (4.1) and by resolving the total cutting forces in three
mutually perpendicular directions, the cutting force components in oblique cutting
can be derived as shown by:
cwcecsc FFFF ++= (4.24)
twtetst FFFF ++= (4.25)
twrersr FFFF ++= (4.26)
By substituting the relevant equations for the forces from the three sources, equations
for the cutting force in oblique cutting are given by:
)cos(/ iVBbCbCbtKF cwcecsc ⋅++= (4.27)
)cos(/)cos(/ iVBbCibCbtKF twtetst ⋅++= (4.28)
)tan(ibCbtKF cersr += (4.29)
Thus, by knowing the tool geometries (normal rake angle and inclination angle), cut
geometries (cut thickness and width of cut) and cutting speed, and by using the
cutting quantities database and the intensities of the edge force and wearland force in
orthogonal cutting, the three cutting force components can be predicted. The
procedure for implementing the cutting force models is given in the flowchart shown
in Figure 4.5. The models will be experimentally verified in the next section.
102
START
Input: Tool insert type and work material
Input: Width of cut b, Cut thickness t, Cutting speed, Rake angle γn, Inclination angle i, Wearland size VB
Obtain τ, β, rl, Cce, Cte, Ccw and Ctw from Database
Calculate:Shear forces Fcs, Fts, FrsEdge forces Fce, Fte, Fre
Flank wearland forces Fcw, Ftw
The total forces Fc, Ft, Fr
Calculate φn, βn ηc
Output: Three predicted force components Fc, Ft and Fr
STOP
Yes
No
Is the cutting condition changedfor a new cut?
Figure 4.5 Procedure for cutting force prediction in oblique cutting.
4.2 Model verification In order to verify the models and to estimate the overall predictive capability of the
models, which are developed for oblique cutting with the effects of the tool
wearland, it is necessary to assess the models qualitatively and quantitatively with
respect to the experimental results. The appropriateness of the database obtained
from the orthogonal cutting tests will be also justified for the prediction of the cutting
forces.
4.2.1 Experimental work A cutting experiment was conducted in order to assess the plausibility and predictive
capability of the oblique cutting force models. The experiment will enable
103
comparisons to be made between the predicted and experimentally measured forces
so that the effects of the cutting variables, such as cut thickness t, wearland size VB,
normal rake angle γn, and inclination angle i, on the predicted forces will be analysed
qualitatively and quantitatively.
Major equipment and experimental procedures used were the same as those in the
orthogonal cutting experiment. The Leadwell CNC lathe was used to cut a mild
carbon steel CS1020 using grade TP20 carbide inserts. The inserts had a flat rake
face with an 8 µm TiN coating. The main chemical composition of the work material
is 0.2% C, 0.6% Mn, 0.06% P and 0.06% S, and its tensile strength is 380 MPa and
hardness is 130 BHN. The specimens were prepared in the tubular shape with a wall
thickness of 3 mm and machined from an end.
The cut thickness, normal rake angle, inclination angle and cutting speed have been
selected such that they cover the practice range of applications and are within the
limits of the equipment. Specifically, three levels of cut thickness (0.1, 0.17 and
0.24mm), three levels of normal rake angle (-5°, 0° and 5°) and three levels of
inclination angle (-5o, 0 and 5o) were tested at two levels of cutting speed (100m/min
and 200m/min). When the inclination angle is zero, the oblique cutting becomes an
orthogonal cutting, so that the cutting forces are identical to those obtained in
orthogonal cutting. As such, the test data for zero inclination were taken from the
orthogonal cutting tests. In addition, three levels of wearland size (0.2, 0.4 and 0.6
mm) were selected as was done in the orthogonal cutting tests. The tool wearland
was artificially made by a lapping process and checked under a shadowgraph
projector to ensure that its size was within 3% of the specified size. Table 4.1 gives
the experimental design specification. Thus, the cutting forces from 162 tests were
obtained for model verification.
Table 4.1 Parameters used in oblique cutting tests
Cut thickness t (mm) 0.1 0.17 0.24 Cutting speed V (m/min) 100 200
Inclination angle i (degree) -5 0 5 Normal rake angle γn (degree) -5° 0° 5°
Wearland VB (mm) 0.2 0.4 0.6
104
The forces signals were measured by a Kistler type 9257A dynamometer that was
mounted on the tool post with a specially made rest, and the tool was mounted on the
dynamometer. The measured force signals were then amplified by three Kistler type
5001 charge amplifiers and recorded by computer with an A/D converter and data
acquisition software. The final result for each force component was the average value
of 40 force samples in the steady cutting stage for each cut. The original
experimental data is listed in Appendix B.
4.2.2 Results and discussion In order to assess the model prediction, the cutting forces for the same cutting
conditions as those used in the experiment in the last section were calculated using
the developed oblique cutting force models. An analysis of the characteristics of the
predicted forces by the oblique cutting force models with respect to the process
parameters has been carried out firstly to study the model’s generality and
plausibility. Qualitative comparisons of the predicted forces are made by plotting the
trends of the predicted forces together with measured forces with respect to cut
variables. Quantitative comparisons are made by analysing the statistical deviations
of the predicted forces with the experimental forces under corresponding cutting
conditions. Figures 4.7 to 4.10 show the trends of the predicted forces by solid lines
while the symbols denote the measured experimental forces.
The typical trends of the cutting forces with respect to cut thickness are shown in
Figures 4.6(a) and (b) for different cutting conditions. All the three force components
Fc , Ft and Fr are shown to linearly increase with cut thickness t, as expected. An
increase in the cut thickness increases the cutting area and hence the cutting forces.
The models have correctly predicted this trend which is consistent with the
experimental trends of the cutting force components. It can also be seen that there is
a positive intercept for each force component at the force axis as a result of the edge
and wearland forces. Qualitatively, the measured experimental forces agreed with the
predicted values very well under the corresponding cutting conditions. It is evidence
from the figures that the models give an excellent prediction of the cutting forces in
oblique cutting.
105
0
500
1000
1500
2000
0 0.1 0.2 0.3Cut thickness t(mm)
F (N
)
Fcm (N)Ftm (N)Frm (N)Predicted
V=100m/min i=5degγ n=0degVB=0.2mm
Fc
Ft
Fr0
500
1000
1500
2000
0 0.1 0.2 0.3Cut thickness t(mm)
F (N
)
Fcm (N)Ftm (N)Frm (N)Predicted
V=200m/min i=5degγ n=5degVB=0.2mm
Fc
Fc
Ft
Fr
(a) (b)
Figure 4.6 Predicted and experimental forces vs. cut thickness.
0
500
1000
1500
2000
0 0.2 0.4 0.6 0.8VB (mm)
F (N
)
Fcm (N)Ftm (N)Frm (N)Predicted
V=100m/min i=5degγ n=5deg t=0.1mm
Fc
Ft
Fr
0
500
1000
1500
2000
0 0.2 0.4 0.6 0.8VB (mm)
F (N
)
Fcm (N)Ftm (N)Frm (N)Predicted
V=200m/min i=5degγ n= -5deg t=0.1mm
Fc
Ft
Fr
(a) (b)
Figure 4.7 Predicted and experimental forces vs. wearland size.
The trends of the cutting force components (Fc, Ft and Fr) with respect to wearland
size are shown in Figures 4.6(a) and (b), from which the influence of the wearland
size on the predicted and measured forces are evident.
The predicted cutting and thrust forces increase linearly with the wearland size as
shown in the Figures 4.7(a) and (b) and align with the trends of the experimental
forces. This is because the wearland size is directly proportional to wearland force
and hence increases the overall cutting forces. The predicted force values agree with
the experimental data from the cutting tests. It can also be seen that the radial force is
independent from the wearland size, as the wearland force was resolved in the
cutting direction and thrust direction only. The experimental radial forces have
confirmed that the models have been correctly developed, while the predicted radial
106
forces in the figures have confirmed that the models have been correctly
programmed and implemented. The characteristics of the power and thrust force
component may form the basis for developing condition monitoring strategies for
sensing imminent tool failure in oblique cutting.
Figures 4.8(a) and (b) show the trends of the predicted and measured forces with
normal rake angle. The Fc and Ft force components decrease with an increase in the
normal rake angle. The Fr component shows a little decrease with the normal rake
angle. It can be seen that the model predictions agree with the trends from the
mechanics of cutting analysis and agree very well with the experimental results.
0
500
1000
1500
2000
-10 -5 0 5 10Normal rake angle γn (degree)
F (N
)
FcmFtmFrmPredicted
V=100m/min i=5degVB=0.2mm t=0.1mm/r
Fc
Ft
Fr0
500
1000
1500
2000
-10 -5 0 5 10Normal rake angle γn (degree)
F (N
)
FcmFtmFrmPredicted
V=200m/min i=5degVB=0.2mm t=0.17mm/r
Fc
Ft
Fr
(a) (b)
Figure 4.8 Predicted and measured forces for different normal rake angle.
The effect of cutting speed on the predicted cutting force components can be found
from Figure 4.9 where the solid lines stand for cutting speed at 200m/min and dashed
lines stand speed at 100m/min. An increase in cutting speed results in very slight
decreases in cutting forces. It can be seen from the figures that the predicted cutting
forces for both cutting speeds are very close, particularly for the cutting and radial
forces. Examining the measured forces with respect to cutting speed shows a similar
trend where cutting speed has minimal effect on the forces. Although the predicted
values are slightly higher than the experimental data, the model predictions are still
excellent both qualitatively in terms of the force characteristics and quantitatively.
The minimal variations of the forces in the Figures 4.8 and 4.9 indicate the cutting
speed has a minimal effect on the force components, because this is in the basic
cutting quantities database from orthogonal cutting.
107
0
500
1000
1500
2000
0 0.1 0.2 0.3Cut thickness t(mm)
F (N
)
V=100m/minV=200m/minSeries2
VB=0.4mmi= 5degγn=5deg
Fc
Ft
Fr
PredictedV=100m/min
PredictedV=200m/min
0
500
1000
1500
2000
0 0.1 0.2 0.3Cut thickness t(mm)
F (N
)
V=100m/minV=200m/minSeries2
VB=0.2mmi= -5degγn=0deg
Fc
Ft
Fr
PredictedV=100m/min PredictedV=200m/min
(a) (b)
Figure 4.9 Predicted and measured forces vs cut thickness for different speeds.
The predicted trends of cutting forces with respect to inclination angle are shown in
Figure 4.10. From the figure, it is found that the inclination angle has a very light
effect on the cutting forces, possibly because of the small increment of the angle
selected in the cutting tests and calculations. The negative radial force at the
inclination angle for -5o occurs, because the negative inclination angle has changed
the direction of the radial force. The model has correctly predicted this trend which is
in good agreement with the experimental data. For all the force components, the
predicted force components agree very well with the measured forces from the
cutting tests. This again shows that the developed models can provide adequate
predictions of cutting force in oblique cutting.
-500
0
500
1000
1500
2000
-10 -5 0 5 10
Inclination angle i (degree)
F (N
)
FcmFtmFrm
FcFt
Fr
PredictedVB=0.2mm γn=0deg.V=100m/min t=0.1mm/r
-500
0
500
1000
1500
2000
-10 -5 0 5 10
Inclination angle i (degree)
F (N
)
FcmFtmFrm
Fc
Ft
Fr
Predicted
VB=0.6mm γn= -5deg.V=200m/min t=0.17mm/r
(a) (b)
Figure 4.10 Predicted and measured forces for inclination angles.
For a quantitative assessment of the predictive capability of cutting force models, the
percentage deviations of the model predicted values with respect to the
108
corresponding experimental data have been analyzed using the same equation used in
Chapter 3. The histograms of percentage deviation of the predicted cutting forces
with respect to the experimental data are given in Figures 4.11 to 4.14.
Figures 4.11(a) to (c) show the percentage deviations of the power force components
at wearland size of 0.2mm, 0.4mm and 0.6mm. The mean values of the percentage
deviations are 0.9%, 3.7% and 5.3% for a wearland size of 0.2mm, 0.4mm and
0.6mm, respectively. The corresponding standard deviations are 3.99%, 5.14% and
7.7%. These percentage deviations show an excellent correlation between the
predicted and experimental cutting force components.
Figures 4.12(a) to (c) show the percentage deviations of the thrust force component
Ft for different wearland sizes. The mean deviations for Ft are -2.4% for VB=0.2mm,
1.0% for VB=0.4mm and 2.4% for VB=0.6mm with standard deviations of 8.23%,
6.72%, and 8.1% respectively for the three wearland sizes. It again shows that the
models gave excellent predictions for the thrust force component.
Figures 4.13(a) to (c) show the percentage deviations of the predicted radial force
from the experimental radial force under the corresponding cutting conditions. It can
be been seen that the mean deviations for the thrust force component are larger than
those for the other two force components, so are the standard deviations for the radial
force. It is not difficult to understand from the predicted and measured values of the
thrust force that all the radial forces are less than 105N, which are significantly
smaller than the power and thrust force components. Therefore, a small error in the
prediction or experiments for the radial force will result in a large percentage
deviation. The relatively larger deviation of the predicted radial force from the
corresponding experimental value in fact represents a small absolute difference in the
force values.
In order to assess the adequacy of the predictive models, a further comparison was
carried out for all the cutting conditions under consideration. This is shown in
Figures 4.14(a) to (c). These comparisons show that the model’s predictions yield an
average percentage deviation of 3.3% for power force, 0.3% for thrust force and
13.3% for radial force, while the corresponding standard percentage deviations are
109
6.04%, 8.1% and 28.89% respectively. It appears the percentage deviation (28.89%)
for radial force is larger than other forces due to the following reasons: the main
reason is the values of the measured and calculated radial forces which are less than
100(N) and mostly around 50(N). If there is a small variation in the radial force, this
will result in a larger percentage deviation. In comparison, the values of cutting force
and thrust force are much higher than the radial force. Consequently, the adequacy of
the predictive cutting force models for oblique cutting with tool flank wear has been
amply demonstrated.
110
Mean=0.9%Std. Dev=3.99%
-45 -30 -15 0 15 30 45
Percentage deviation (Fc) VB=0.2mm
0%
20%
40%
60%
Freq
uenc
y %
(a)
Mean=3.7%Std. Dev=5.14%
-45 -30 -15 0 15 30 45
Percentage deviation (Fc) VB=0.4mm
0%
20%
40%
60%
Freq
uenc
y %
(b)
Mean=5.3%Std.Dev=7.7%
-45 -30 -15 0 15 30 45
Percentage deviation (Fc) VB=0.6mm
0%
20%
40%
60%
Freq
uenc
y %
(c)
Figure 4.11 Percentage deviations between predicted and experimental power force components for different VB
111
Mean= -2.4%Std. Dev=8.23%
-45 -30 -15 0 15 30 45
Percentage deviatoin (Ft) VB=0.2mm
0%
20%
40%
60%
Freq
uenc
y %
(a)
Mean=1%Std. Dev=6.72%
-45 -30 -15 0 15 30 45
Percentage deviatoin (Ft) VB=0.4mm
0%
20%
40%
60%
Freq
uenc
y %
(b)
Mean=2.4%Std. Dev=8.1%
-45 -30 -15 0 15 30 45
Percentage deviation (Ft) VB=0.6mm
0%
20%
40%
60%
Freq
uenc
y %
(c)
Figure 4.12 Percentage deviations between predicted and experimental thrust force components for different VB
112
Mean=18.3%Std. Dev=27.07%
-45 -30 -15 0 15 30 45
Percentage deviation (Fr) VB=0.2mm
0%
20%
40%
60%
Freq
uenc
y %
(a)
Mean=15.2%Std.Dev=30.44%
-45 -30 -15 0 15 30 45
Percentage deviation (Fr) VB=0.4mm
0%
20%
40%
60%
Freq
uenc
y %
(b)
Mean=6.4%Std.Dev=31.55%
-45 -30 -15 0 15 30 45
Percentage deviation (Fr) VB=0.6mm
0%
20%
40%
60%
Freq
uenc
y %
(c)
Figure 4.13 Percentage deviations between predicted and experimental radial force components for different VB
113
Mean=3.3%Std.Dev=6.04%
-45 -30 -15 0 15 30 45
Percentage deviation (Fc)
0%
20%
40%
60%
Freq
uenc
y %
(a)
Mean=0.3%Std.Dev=7.9%
-45 -30 -15 0 15 30 45
Percentage deviation (Ft)
0%
20%
40%
60%
Freq
uenc
y %
(b)
Mean=13.3%Std.Dev=28.89%
-45 -30 -15 0 15 30 45
Percentage deviation (Fr)
0%
20%
40%
60%
Freq
uenc
y %
(c)
Figure 4.14 Percentage deviations between predicted and experimental cutting forces.
114
4.3 Concluding remarks Predictive cutting force models for oblique cutting with tool flank wear has been
developed in this chapter. The models were an extension of the mechanics of cutting
analysis for orthogonal cutting with tool flank wear which was presented in the
previous chapter. The forces required to form the chips were modelled using the
popular thin shear zone analysis while taking the basic cutting quantities developed
for orthogonal cutting. It is important to note that with this modelling approach, only
orthogonal cutting tests are needed to determine the basic cutting quantities. The
models for oblique cutting and other practical machining operations can then be
developed. The analysis of edge force was taken from the literature and reconsidered
when being applied to oblique cutting with tool flank wear. An analysis of the forces
generated on the wealand has also been carried out. This produced the wearland
force models for oblique cutting. Based on the analysis in orthogonal cutting, the
overall cutting forces for oblique cutting were a result of the force components from
all three sources, i.e. the force for basic chip formation, the edge force and wearland
force.
A model verification experiment was carried out which covered a wide range of
process variables relevant to common applications. The three cutting force
components for each test were acquired to assess the predictive models. The cutting
forces were also evaluated by the predictive cutting force models under cutting
conditions corresponding to the experiment, thus enabling comparison of the
predicted and experimental cutting forces to be made.
Qualitative assessment of the models was carried out by analysing the characteristics
of the predicted forces and by comparing with these corresponding experimental
data. It has been shown that the model predictions are plausible and accurately reflect
the effects of the different process variables. It has also been shown that predicted
force trends correlate very well with the corresponding experimental data.
Quantitative assessment was made based on the percentage deviation of the predicted
forces from the corresponding experimental forces and by statistically analysing the
average and standard deviations with the assistance of histograms. The analysis
115
showed that the cutting force models can give accurate predictions under different
wearland sizes. The average and standard deviations of the cutting forces for the
overall cutting conditions under consideration showed again that the models can
yield adequate prediction of the cutting forces in oblique cutting with tool flank
wear.
In the next chapter, the cutting force models and modelling approach for orthogonal
and oblique cuttings will be extended to a practical machining operation, i.e. turning
operation.
116
Chapter 5 Predictive cutting force models for turning operations allowing for the effects of flank
wear
Turning operation is a common cutting process. Although the basic cutting action and
phenomena in turning operations are similar to those in the classic orthogonal and
oblique cutting processes, the cutting tools used in the turning operations are more
complex [24]. This complexity comes from the complex tool geometries which
include an approach angle and end cutting edge angle, in addition to rake angle and
inclination angle, as well as the complicated edge shape, i.e. the cutting edge includes
a straight major (or side) cutting edge, a straight minor (or end) cutting edge and often
a nose radius edge. In addition, when a nose radius tool is used in turning operations,
the cut thickness is variable which further complicates the cutting process and makes
the modelling work difficult.
In this chapter, the cutting force model for turning operations will be developed
incorporating the force required for chip formation, the edge force and the wearland
force. For the purpose of the modelling work, attention is paid to only the more
common cutting process where a nose radius lathe tool is used. The concept of an
equivalent cutting edge [23, 67, 113] will be used in order to apply the machining
theories to turning operations. The forces required for basic chip formation will be
modelled on the equivalent cutting edge using the thin shear zone mechanics of
cutting analysis. The edge force will be reconsidered and modelled along the actual
cutting edge in the turning operation based on the work in Chapters 3 and 4.
Similarly, the wearland force will be analysed and modelled for the turning operation
according to the relevant contact area and the directions of rubbing forces. The
plausibility of the model will then be studied by analysing the predicted trends. The
model will finally be verified by an experimental investigation.
5.1 Overview of the modelling approach
The ‘Tool-in-hand System’ and the reference planes with respect to points on the tool
cutting edges will be used in the study, which is defined by the ISO 3002 [114] to
specify the geometry of the tool cutting part on a point to point basis. In Australia,
117
the ‘Tool-Design System’ has been proposed which describes the tool geometry with
the common straight shanked tool in the conventional operating position, i.e. with a
point on the cutting edge set ‘on centre’ and the tool base plane perpendicular to the
assumed direction of primary motion and the tool axis mutually perpendicular to the
assumed directions of primary and feed motions (Australian Standards 1979 [115]).
The ISO and Australian System are the same when the tools under the ISO standard
are also set in this conventional operating position. For this ‘static’ situation, the
assumed directions of primary and feed motions are fixed for all points on the cutting
edges. Furthermore, for points on the straight major and minor cutting edges, the
corresponding angles at different positions on the cutting edges are the same, e.g. the
approach angle Cs has the same value for all points on the straight major cutting
edge, showing in Figure 5.1. By contrast, many of the planes and angles on points on
the curved cutting edge at the ‘corner’ may change from point to point.
Cs
γο
Pf
i
γn
Ce
M
M
M-M
N
N
N-N
Figure 5.1 Single-point tool angles
In order to develop the cutting force model, the following assumptions have been
made:
1. A continuous chip is formed without a built-up edge.
2. The tool is not perfectly sharp. There are concentrated edge forces acting on the
cutting edge and uniformly distributed along the cutting edge.
118
3. The rubbing (or wearland) force acts on the wearland at the tool flank, and is
uniformly distributed on all the contact area on the wearland.
4. The shearing process occurs in a localized (thin) shear zone, idealized by a
plane, extending from the cutting edge to the work surface. According to the
study in Chapter 3, the forces on the shear zone are not affected by tool flank
wear.
5. The shear stress on the shear plane is uniformly distributed.
6. The collinearity conditions apply. The friction force F on the rake face is
collinear to the chip flow direction. The shear force Fs in the shear plane is
collinear to the shear velocity and the wearland force Fw is collinear to the
cutting direction.
Fys
Fxs
Fzs
Fxw
Fzw
Fxe
FzeFyw
Fye
VB f
d
Figure 5.2 Forces in general turning operation.
If the tool nose radius edge is engaged in cutting in a turning operation, the un-cut
chip thickness is uneven as shown in Figure 5.2 and in references [23, 67, 113].
Based on the study in Chapters 3 and 4, the total cutting force is a result of the forces
from three sources, namely, the forces required for basic chip formation, edge forces
on the cutting edge, and rubbing forces on the wearland (or wearland force).
Mathematically, the total cutting force can be expressed as:
F = Fs + Fe + Fw (5.1)
119
Each of the three forces can be projected on the feed, radial and cutting speed
directions, denoted by Fx, Fy and Fz respectively. Accordingly, the three force
components for each of the three force sources are: Fxs, Fys, and Fzs for the forces
required for the basic chip formation, Fxe, Fye, and Fze for the edge forces, and Fxw,
Fyw, and Fzw for the wearland forces.
dFY
dFX
dFZ
dFXw
dFZw
dFXe
dFZedFYw
dFYe
ΩΩΩΩ i
dF O
VB
Figure 5.3 Elemental forces in a general turning operation.
The predictive cutting force models allowing for the flank wear in turning operations
that will be presented in the next sections will be developed in the following stages:
• The equivalent cutting edge will first be determined using the concept and
models given in [23, 67, 68, 113] for the nose radius tools. With the equivalent
cutting edge, a tool with a nose radius in turning operations can be treated as if
it were a tool with a single straight cutting edge (equivalent cutting edge) using
the modified cutting edge angle with reference to the equivalent cutting edge.
The force for chip formation will be determined by applying the machining
theories to the equivalent cutting edge in the same way as that in the oblique
cutting presented in Chapter 4.
• The edge forces will be determined by integrating all the forces on each
infinitesimal element along the cutting edge engaged in cutting using the edge
force intensity factors established in Chapter 3, as shown in Figure 5.3.
120
• The wearland forces will be determined by integrating the forces on a series of
elements of infinitesimal width on the wearland as shown in the Figure 5.3.
• The total cutting forces involved in a turning operation is the sum of the forces
from the above three sources.
5.2 Forces for basic chip formation in turning operation The forces required for the basic chip formation in turning operations can be
represented by the tangential force component, Fzs, the feed force component, Fxs,
and the radial force component Fys. The tangential force controls the power and
torque of the cutting. The feed force is parallel to the tool transition and therefore it
controls the load acting on the feed mechanism. The radial force induces deflections
on the tool holder and workpiece in the relevant direction thereby controlling the
accuracy of the finished component. If the cutting conditions are Vf<<V, d<<D and
f<<d, and the tool nose radius and end cutting edge (if engaged in cutting) are
ignored in the cutting analysis, the turning operation is similar to the classical
orthogonal or oblique cutting processes at the side cutting edge. However, lathe tools
used in practice often have a round corner which changes the cut thickness and chip
flow, and its effects cannot be ignored, especially in a high f/d ratio, such as in finish
cutting where the entire cutting edge engaged in cutting may be on the nose radius.
As specified in many tool manufacturers’ brochures, the end cutting should not be
engaged in cutting, although this may be not the case in some machining operations.
Nevertheless, this study is limited to the cutting where end cutting edge is not
engaged in cutting. To consider the effect of nose radius edge in predicting the forces
required for basic chip formation, the equivalent cutting edge concept [23, 67, 68,
113] will be used, as presented below.
5.2.1 The equivalent cutting edge concept
As described in the literature review, equivalent cutting edge appears to be a popular
and effective means for predicting the cutting performance in practical operations
with nose radius tools under various cutting conditions. Several approaches were
proposed to define and determine the equivalent cutting edge in turning operations
121
over the last decades. Hu et al. [65] proposed an equivalent cutting edge for lathe
tools with two straight cutting edges and a sharp corner (no nose radius) under
orthogonal cutting conditions (with zero inclination and rake angles). The equivalent
cutting edge was defined to be the line joining the two intersection points of the side
cutting edge with the uncut work surface and the end cutting edge with the newly cut
spiral work surface in the rake of the tool.
For nose radius tools, Colwell [62] suggested a major axis of the projected area by
geometrical analysis to determine the equivalent cutting edge. The major axis is in
the segment joining the extreme joints of the engaged cutting edge for a tool with a
zero tool inclination angle and rake angle. Young et al. [22] developed a
mathematical model to determine equivalent cutting for predicting the chip flow
angle for a nose radius tool. In this work, it was assumed that the elemental friction
force at the cutting area was proportional to the local undeformed chip thickness and
collinear with the local chip velocity. The elemental friction forces were resolved to
obtain the resultant friction force and its direction. It was assumed that the friction
force direction would coincide with the chip flow direction. However, this study was
limited to zero inclination angle and zero rake angle.
In order to consider general practical tools, Wang et al. [23, 67, 68] has further
developed Young’s work and proposed a new equivalent cutting edge concept in
predicting the cutting performance in turning operations with nose radius tools under
oblique cutting conditions. In this work, the authors separately consider the effect of
nose radius edge (and end cutting edge) and that of the oblique cutting conditions.
The chip flow direction (or the deviation from the normal to the major cutting edge)
caused by the nose radius edge is determined when zero inclination and zero rake
angles were considered. The chip flow direction is assumed to be in the direction of
the resultant friction force between the chip and tool rake, and the equivalent cutting
edge was defined as an imaginary line that lies in the tool rake face, passes the
joining point of the side cutting edge and the uncut workpiece surface and is normal
to the chip flow direction. Once the equivalent cutting edge is determined, the
equivalent tool angles with reference to the equivalent cutting edge are then
determined and used with the depth of cut and feed rate to predict the cutting forces
using a machining theory such as the variable flow stress theory [68]. This equivalent
122
cutting edge concept will be used in this study and associated models developed in
the following sections, as showing in Figure 5.4.
Figure 5.4 Equivalent cutting edge for nose radius tool [23]
5.2.2 The chip flow angle caused by tool nose radius and the equivalent cutting edge
It has been suggested that the friction force on the chip at the tool-chip interface is
the sole factor in affecting the chip flow direction so that the chip flow direction is
the same as that of the friction force [81]. In the early studies [78, 81] the chip was
treated as a series of elements of infinitesimal width, each having its own thickness
and orientation. The magnitude of the friction force on the element was directly
proportional to the local undeformed chip thickness and the direction was coincided
with the local chip velocity. The entire chip flow direction was considered to take the
direction of the resultant friction force on the chip. In determining the equivalent
cutting edge, the inclination angle and rake angle are initially not considered [23, 67,
68]. Thus the friction force direction on each element is perpendicular to the local
cutting edge, irrespective of the effect of the inclination of the local cutting edge. The
chip flow model used for determining the chip flow angle caused by the nose radius
edge is taken from [68] as shown in Figure 5.5.
123
dFoY
dFoX
ΩΩΩΩ0000
dFoCs
Y
XZ
Figure 5.5 General chip flow model on rake face
(Note: The X and Y axes lie in the rake face plane.)
The force components were assumed to be on the rake face plane. Based on this
model, the magnitude of the friction force |dF0| acting on an arbitrary small chip
element can be expressed by:
udAdF =|| 0 (5.2)
Where: u (N/mm2) is friction force intensity;
dA (mm2) is the area of the small chip element.
The differential friction force was resolved into two components in the X and Y
directions as shown in Figure. 5.3:
00 sin|||| Ω= dFdFox (5.3)
00 sin|||| Ω= dFdFoy (5.4)
Where: 0 (degree) is the angle made by dFo with the positive Y axis and was given
by:
θπ −=Ω2
)(s (5.5)
Where: θ (degree) is the angle made by positive X axis with the undeformed chip
element. The integral forms of the equations (5.3 and 5.4) are given:
124
Ω= dAuFox 0sin (5.6)
Ω= dAuFoy 0cos (5.7)
As the chip velocity is assumed to be coincidental with the friction force vector, the
angle made by the resultant friction force Fo and the positive Y-axis can be
determined by:
Ω
Ω=Ω
−
dA
dA
0
010
cos
sintan (5.8)
This is the fundamental equation for the angle of 0Ω , from which the chip flow angle
with reference to the major or side cutting edge can be derived. It can be seen that the
cutting force intensity u is eliminated from the equation, so that the chip flow
direction can be predicted by the tool and cutting geometries only.
As in references [23, 68], there may be four different possible uncut chip sections
depending on the possible combinations of the feed rate (f) and depths of cut (d),
approach angle (Cs), end cutting edge angle and nose radius (r). However, according
to the ISO standard on single point cutting tools [70], the maximum feed is limited to
less than 0.8 times the corner radius (i.e. rf 8.0≤ ) so that the end (or minor) cutting
edge cannot be engaged in cutting. Thus, only two cases are possible in practice as in
Wang’s work [113], as shown in Figures 5.4 and 5.5.
In the following sections, the models for the equivalent cutting edge will be
developed based on the work of Wang et al. [23, 68, 113]. Because of the similarity
of this study and the previous work in determining the equivalent cutting edge in
turning operations, detailed derivations will not be presented; but the final and
essential models required for predicting the cutting forces will. The expressions for
the numerator (NUM) and denominator (DEN) for equation (5.17) for the two
possible cases will be given below.
125
Case 1. Cutting conditions: )sin1( ''sCrd −>
The r is the tool nose radius. d’ and C’s are the projection of depth of cut d and
approach angle Cs on the rake face, respectively, and are given by:
2/1222' ]seccos)sintan[(tanseccos ssns CiiCiCdd ++= γ (5.9)
( )[ ]
++
+= −2/1222
1'
seccossintantan
tantansin1cos
ssn
nss
CiiC
CiC
γγ
(5.10)
O
f/2
1
rY
XO
f/2
d’
2
C’s
A
B
O’C’s
r(1-sinC’s)
Figure 5.6 Geometrical relation of chip flow (Case 1).
In dealing with the uncut chip section, an approximation was made in region A
where the inner arc in fact contains a small portion of straight cutting edge. This
approximation was found to produce less than 0.1% error in the final chip flow
direction [23, 68, 113].
By integrating the elemental chip sections in a way similar to [68], the expressions
for the numerator (NUM) and denominator (DEN) for areas A and B are respectively
given by:
126
2
1)]sin(sin)sin(sin
)]2sin(21
[21
sin[
12
21
222 θθθθθ
θθθ
rf
fr
fr
frrNUM A
−+−+
++−= (5.11)
2
1])sin(coslog
21
)sin(cos21
)]2cos(41
cos[
21
22222
21
222
θθθθ
θθθθ
frff
fr
frrrDENA
−+−+
−++−= (5.12)
BsB ACNUM 'cos= (5.13)
BsB ACDEN 'sin= (5.14)
Where:
)]2sin(41
)sin1([ '''ssB CfCrdfA −−−= (5.15)
The corresponding limits of integration in this case are
)2
(cos 11 r
f−=θ (5.16)
'2 sC−= πθ (5.17)
O
f/2
1
2
Y
XO
f/2
d’
3
A
B
O’
r-d’
r
Figure 5.7 Geometrical relations of chip flow (Case 2).
127
Case 2. Cutting conditions: )sin1( ''sCrd −≤
As shown in Figure 5.7, the uncut chip section for this case is again divided into two
regions. The expressions for the corresponding terms of the numerator and
denominator for region A are the same as in Case 1, i.e.
2
1)]sin(sin)sin(sin
)]2sin(21
[21
sin[
12
21
222 θθθθθ
θθθ
rf
fr
fr
frrNUM A
−+−+
++−= (c.f. 5.11)
2
1])sin(coslog
21
)sin(cos21
)]2cos(41
cos[
21
22222
21
222
θθθθ
θθθθ
frff
fr
frrrDENA
−+−+
−++−= (c.f. 5.12)
While the limits of the integration are given by:
)2
(cos 11 r
f−=θ (c.f. 5.16)
−−
−−= −
fdrd
dr
21
2''
'1
2
)2(tanπθ (5.18)
The corresponding terms of the numerator and denominator for the region B are
given by:
3
2]sin)log(sin)[( ' θ
θθθ rdrrNUM B −−= (5.19)
3
2])(cos[ ' θ
θθθ drrrDENB −+−= (5.20)
Where: the limit of the integration for 3θ is given by:
)(sin'
13 r
dr −−= −πθ (5.21)
128
Consequently, the angle of chip flow with reference to the Y axis, 0Ω , for the two
cases can be given in the general form of:
ΣΣ
=Ω −
j
j
DEN
NUM10 tan (j = A, B) (5.22)
The angle, η0, caused by the nose radius edge and measured from the normal to the
straight part of the side cutting edge in the rake face plane is determined from:
0'
0 2Ω−−= SC
πη (5.23)
Deriving above equations, the feed f was used, rather than its projection in the rake
face. For the small rake and inclination angles used in this study, this simplification
will not cause any discernible errors in evaluating the chip flow direction.
From the above analysis, if the tool geometry, γn, i, Cs and r, together with cutting
conditions, f and d are given, it is possible to calculate the chip flow angle η0 in the
rake face. The equivalent cutting edge can then be determined as an imaginary line
on the tool rake face that passes the intersection of the side cutting edge and the
uncut workpiece surface and is normal to the chip flow direction.
5.2.3 Equivalent tool angles for the equivalent cutting edge
Five basic planes are needed to define the angles at the selected point on the
equivalent cutting edge according to the ISO standard for a single-point tool. When
the basic planes are applied to the equivalent cutting edge, a star superscript (*) is
superscribed to the corresponding symbol, i.e. P*s is the tool cutting edge plane,
which is tangential to the equivalent cutting edge and normal to the tool reference
plane, P*n is the cutting edge normal plane, which is perpendicular to the equivalent
cutting edge. The other basic planes, reference plane Pr (parallel to the tool base),
tool back plane Pp (perpendicular to the axis of the work) and assumed work plane Pf
(parallel to the direction of feed motion) remain the same. The modified tool angles
are defined as:
129
• Inclination angle, i*, which is the angle measured in P*s, between the equivalent
cutting edge and Pr and is positive if the equivalent cutting edge is sloping
downwards from the end nearer to the axis of work.
• Normal rake angle, γ*n, which is the angle measured in P*
n, made by the
equivalent cutting edge with the rake face plane and is positive if the rake face
is sloping downwards in the outward direction with respect to the equivalent
cutting edge.
• The approach angle, C*s, is the acute angle that P*
s makes with Pp and is
measured in the reference plane Pr. It is positive in the clockwise sense.
The angles i*, γ*n and C*
s for the equivalent cutting edge are determined by a three-
dimensional geometrical analysis, which is given by [23, 68]:
)cossinsinsin(cossin* 001 iii nγηη −= − (5.24)
)costan
sinsinsec(sin *
0
*01*
iii
n ηηγ −= − (5.25)
and
'0
* η+= ss CC (5.26)
Where: η’0 is the projection of η0 in the reference plane Pr.
η’0 is given by:
)cos
sinsinsincoscos(cos *
001'0 i
ii nγηηη += − (5.27)
5.2.4 The forces for basic chip formation
Turning operations using a single-point lathe tool with a nose radius edge has been
converted to an oblique cutting process using an equivalent straight cutting edge, as
described in the previous sections. It is now possible to use the oblique cutting model
developed in Chapter 4, together with the basic cutting quantity database developed
in Chapter 3, to predict the forces required for basic chip formation when a nose
radius lathe tool is used (note that orthogonal cutting is a special case of oblique
130
cutting, so that the oblique cutting model can be used to deal with orthogonal
cutting). It should be emphasized that the relevant tool and cut geometries for the
equivalent cutting edge must be used when using the oblique cutting model. The
relevant equations accordingly become:
*** tbKF csc = (5.28)
*** tbKF tst = (5.29)
*** tbKF rsr = (5.30)
Where: coefficients *csK , *
tsK and *rsK (N/mm2) are related to the tool geometries and
cutting conditions for the equivalent cutting edge, described in Chapter 4 are given
by:
*2*2***2
*****
**
sintan)(cos
sintantan)cos(sin
ncnnn
ncnn
ncs
iK
βηγβφβηγβ
φτ
⋅+−+⋅⋅+−⋅= (5.31)
*2*2***2
**
***
sintan)(cos
)sin(cossin
ncnnn
nn
nts i
Kβηγβφ
γβφ
τ⋅+−+
−⋅= (5.32)
*2*2***2
*****
**
sintan)(cos
sintantan)cos(sin
ncnnn
ncnn
nrs
iK
βηγβφβηγβ
φτ
⋅+−+⋅−⋅−⋅= (5.33)
Normal shear angle and friction angle become:
***
****
sin)cos/(cos1cos)cos/(cos
tanncl
ncln ir
irγη
γηφ⋅⋅−
⋅= (5.34)
** costantan cn ηββ ⋅= (5.35)
***
****
tansintancostan
)tan(i
i
nc
nnn ⋅−
⋅=+γη
γβφ (5.36)
It is noted that the equivalent cut thickness (t*) and width of cut (b*) are used in the
force models. From a simple geometrical analysis, it can be found that for a small
nose radius edge and small end cutting edge angle, the product of the equivalent cut
thickness and width of cut is approximately equal to the product of the actual cut
131
thickness (t) and width of cut (b), which can in turn be approximated by the product
of the actual feed per revolution (f) and depth of cut (d).
Consequently, for given values of tool geometry, γn, i, Cs and r, cutting condition, f,
d, and V, together with the basic cutting quantity database, the three components of
cutting forces can be evaluated using the above equations and the same calculation
procedure as given in Chapter 4.
Resolving the three force components (Fc, Ft, and Fr) in the X, Y and Z directions in a
turning operation and replacing t* and b* with t and b, respectively, gives:
dfCKCKF srsstsxs )sincos( **** += (5.37)
dfCKCKF srsstsys )cossin( **** −= (5.38)
dfKF cszs*= (5.39)
Where: Cs* (degree) is the approach angle of the equivalent cutting edge and was
given in equation (5.26).
5.3 Edge forces in turning operations
dFre
dFte
dF’edFye
dFxe
dF’ce
dFce
dFre
Cs
i
iL
db
(dFze)
CsL
iL
Figure 5.8 Edge force analysis in turning operations.
132
The rubbing forces on the cutting edge are shown in the Figure 5.8 in the elemental
form. If the edge element length is db, and the iL and CsL are the local inclination
angle and local approach angle on the chip element under consideration, the three
edge force components for the elemental edge length can be found using the same
analysis as in Chapter 4; namely,
dbCdF cece = (5.40)
dbi
CdF
L
tete cos
= (5.41)
dbiCdF Lcere tan= (5.42)
When the above force components are projected to the X, Y and Z directions in
turning operations, the resulting edge force components in the three directions
become:
sLresLtexe CdFCdFdF sincos += (5.43)
sLresLteye CdFCdFdF cossin −= (5.44)
ceze dFdF = (5.45)
Substituting equations 5.40, 5.41 and 5.42 to 5.43, 5.44 and 5.45 gives:
dbCiCdbCi
CdF sLLcesL
L
texe sintancos
cos+= (5.46)
dbCiCdbCi
CdF sLLcesL
L
teye costansin
cos−= (5.47)
dbCdF ceze = (5.48)
The total edge force components can be obtained by integrating the above equations
along the cutting edge engaged in cutting. Similar to the analysis for the basic chip
formation forces, the cutting edge engaged in cutting may be on the round corner
133
only, or may include a round cornered edge and a straight side cutting edge,
depending on the tool and cut geometries.
Case 1: )sin1( ''sCrd −> - When the straight side cutting edge is engaged in cutting:
When the cutting conditions are such that the straight side cutting edge is engaged in
cutting, the situation is similar to that of Case 1 in the previous section. Thus, the
analysis for edge force will need to be carried out on the straight cutting edge and the
nose radius edge as well.
When considering the straight cutting edge, the local inclination angle is equal to the
tool inclination angle and the local approach angle is equal to the tool approach
angle, i.e. iL=i and CsL=Cs. The edge forces on the straight part of the cutting edge
can be expressed by integrating all the elemental forces along the cutting edge and
can be given by:
straightsceste
xe bCiCCi
CF )sintancos
cos(1 += (5.49)
straightsceste
ye bCiCCi
CF )costansin
cos(1 −= (5.50)
straightceze bCF =1 (5.51)
Where: bstraight (mm) is the length of the straight part of the side cutting edge engaged
in cutting, and can be derived according to the geometrical relationship shown in
Figure 5. 8. Thus, bstraight can be given by:
icosCcos
)Csin1(rdb
's
's
'
straight−−= (5.52)
Where: d’ (mm) and Cs’ (degree) are respectively the projection of depth of cut and
approach angle on the tool rake face and were given in equations (5.9) and (5.10)
above.
134
To evaluate the edge force on the nose radius edge, it is important to determine the
local inclination angle as well as the local approach angle. The local edge inclination
angle on any part of the nose radius edge is taken from reference [113] and is given
by:
[ ])Ccos(isin)Csin(icossinsin)(i 's
'sn
1r +θ++θγ−=θ − (5.53)
Where: (degree) is the angular position of the thickness element counter-clockwise
from the positive X axis as shown in Fig. 5.8, and the other symbols are as defined in
the nomenclature.
The local approach angle is varied along the nose radius edge. From Fig. 5.8, the
projection of the local approach angle on the tool rake plane can be found as:
θπθ −=)('sC (5.54)
For a lathe tool with a nominal inclination angle and rake angle on the major cutting
edge, the inclination angle on the nose radius edge will be small. This is particularly
so when a small inclination angle and a small normal rake angle (normally from -5o
to 5o) for brittle cutting tools, such as carbide and ceramic tools, are used. Thus, the
local approach angle on any part of the nose radius edge can be approximated as:
θπθ −=≈ )('ssr CC (5.55)
The elemental length, db, on the nose radius edge is given by:
θrddb = (5.56)
135
F’te
F’ce
Of
C’s
r
A
B
θ2
θ1
f/2
X
Y
θC’s(θ)
Z
d’
Figure 5.9 Edge forces at nose radius in turning operations.
(Note: the X and Y axes lie in the rake face plane.)
The edge forces on the nose radius edge can then be found by carrying out
integrations on equations (5.46) to (5.48), i.e.:
+=2
1
]sintancoscos
[2
θ
θ
θrdCiCCi
CF srrcesr
r
texe (5.57)
−=2
1
]costansincos
[2
θ
θ
θrdCiCCi
CF srrcesr
r
teye (5.58)
=2
1
2
θ
θ
θrdCF ceze (5.59)
Where: limits of the integration are in two cases as for anglicising chip flow angle in
the previous section:
)2/
(cos 11 r
f−=θ (5.60)
'2 sC−= πθ (5.61)
The total edge forces are the sum of the edge forces on the straight and round edges,
i.e.:
136
21 xexexe FFF += (5.62)
21 yeyeye FFF += (5.63)
21 zezeze FFF += (5.64)
Case 2: )sin1( ''sCrd −≤ - When only the nose radius edge is engaged in cutting:
Under the above condition, the straight side cutting edge will not be engaged in
cutting so that only equations (5.57) to (5.59) apply i.e.
+=2
1
]sintancoscos
[θ
θ
θrdCiCCi
CF srrcesr
r
texe (5.65)
θ−=θ
θ
2
1
rd]CcositanCCsinicos
C[F srrcesr
r
teye (5.66)
θ=θ
θ
2
1
rdCF ceze (5.67)
Where: limits of integrations are:
)2/
(cos 11 r
f−=θ (5.68)
)(sin'
12 r
dr −−= −πθ (5.69)
5.4 Wearland forces in turning operations
The rubbing force on the wearland acts in two directions, one is opposite to the
sliding velocity (or cutting velocity), and the other is normal to the tool wearland. In
analysing the forces, it is assumed that the wearland width is evenly distributed along
the tool flank and the contact area is equal to the wearland area. The wearland forces
can be treated as a series of elements of infinitesimal width, each is in the direction
along the cutting velocity, as shown in Figure 5.10.
137
dFtw
dF’e
dFyw
dFxw
dF’cwdFce
Cs
i db
(dFzw)
CsL
VB
Figure 5.10 Elemental wearland forces in turning operations.
Each of the force elements can be treated as an oblique cutting process. Using the
analysis in Chapter 4, force elements can be expressed by:
dbi
VBCdF
L
cwcw )cos(
= (5.70)
dbi
VBCdF
L
twtw )cos(
= (5.71)
0=rwdF (5.72)
Where: the iL (mm) is the local inclinational angle of the element.
Resolving the elemental wearland force components in the X, Y and Z directions
gives:
twsLxw dFCdF )cos(= (5.73)
twsLyw dFCdF )sin(= (5.74)
138
cwzw dFdF = (5.75)
The total wearland forces can be found according to the tool and cut geometries
which are again categorized into two cases as below.
Case 1: )sin1( ''sCrd −> - When the straight side cutting edge is engaged in cutting:
When both the straight side and nose radius edges are engaged in cutting, the
wearland forces can be evaluated separately on the straight edge and nose radius
edge as follows.
The wearland force on the straight part of side cutting edge can be found from:
straighttw
sxw bi
VBCCF
)cos(cos1 = (5.76)
straighttw
syw bi
VBCCF
)cos(sin1 = (5.77)
straightcw
zw bi
VBCF
)cos(1 = (5.78)
Where: the bstraight (mm) is the length of the straight side cutting edge engaged in
cutting and has been given in the previous section.
Using a similar procedure as in the edge force analysis, the wearland forces on the
nose radius edge are given by (note: θrddb = ):
=2
1)cos(
)cos(2
θ
θ
θrdi
VBCCF
r
twsrxw (5.79)
=2
1)cos(
)sin(2
θ
θ
θrdi
VBCCF
r
twsryw (5.80)
=2
1)cos(2
θ
θ
θrdi
VBCF
r
cwzw (5.81)
139
Where: the limits of integrations are:
)2/
(cos 11 r
f−=θ (5.82)
'2 sC−= πθ (5.83)
The total wearland forces for both the straight and nose radius edges are:
21 xwxwxw FFF += (5.84)
21 ywywyw FFF += (5.85)
21 zwzwzw FFF += (5.86)
Case 2: )sin1( ''sCrd −≤ - When only the nose radius edge is engaged in cutting:
Under this condition, only the nose radius edge is engaged in cutting so that the
wearland forces can be found by taking integrations of the elemental wearland forces
on the nose radius edge engaged in cutting; namely,
θ=θ
θ
2
1
rd)icos(
VBC)Ccos(F
r
twsrxw (5.87)
θ=θ
θ
2
1
rd)icos(
VBC)Csin(F
r
twsryw (5.88)
θ=θ
θ
2
1
rd)icos(
VBCF
r
cwzw (5.89)
Where: limits of integrations are:
)2/
(cos 11 r
f−=θ (5.90)
)(sin'
12 r
dr −−= −πθ (5.91)
140
5.5 Final cutting force model for turning operations allowing for tool flank wear
The cutting forces in turning operations are the sum of the forces for chip formation,
the edge forces and the wearland force as discussed earlier, i.e.
xwxexsx FFFF ++= (5.92)
ywyeysy FFFF ++= (5.93)
zwzezsz FFFF ++= (5.94)
By substituting the relevant force equations from the three sources, the total cutting
force equations are given by:
For case 1 ( )sin1( ''sCrd −> ):
+
+
++
++
+=
2
1
2
1
)cos()cos(
)cos(cos
]sintancoscos
[
)sintancoscos
(
)sincos( ******
θ
θ
θ
θ
θ
θ
rdi
VBCC
bi
VBCC
rdCiCCi
C
bCiCCi
C
tbCKCKF
r
twsr
straightr
tws
srrcesrr
te
straightsceste
srstsx
(5.95)
+
+
−+
−+
−=
2
1
2
1
)cos()sin(
)cos(sin
]costansincos
[
)costansincos
(
)cossin( ******
θ
θ
θ
θ
θ
θ
rdi
VBCC
bi
VBCC
rdCiCCi
C
bCiCCi
C
tbCKCKF
r
twsr
straighttw
s
srrcesrr
te
straightsceste
srsty
(5.96)
141
++++=2
1
2
1)cos()cos(
***θ
θ
θ
θ
θθ rdi
VBCb
iVBC
rdCbCtbKFr
cwstraight
cwcestraightcecsz (5.97)
For case 2 ( )sin1( ''sCrd −≤ :
+
++
+=
2
1
2
1
)cos()cos(
]sintancoscos
[
)sincos( ******
θ
θ
θ
θ
θ
θ
rdi
VBCC
rdCiCCi
C
tbCKCKF
r
twsr
srrcesrr
te
srstsx
(5.98)
+
−+
−=
2
1
2
1
)cos()sin(
]costansincos
[
)cossin( ******
θ
θ
θ
θ
θ
θ
rdi
VBCC
rdCiCCi
C
tbCKCKF
r
twsr
srrcesrr
te
srsty
(5.99)
++=2
1
2
1)cos(
***θ
θ
θ
θ
θθ rdi
VBCrdCtbKF
r
cwcecsz (5.100)
In implementing the above equations, the basic cutting quantities (shear angle,
friction angle, and shear stress) and coefficients of the edge and wearland forces
obtained from the basic cutting quantity database in Chapter 2 are used. A Matlab
program has been written to assist with the calculations. The procedures in using the
force predictive models for turning operations allowing for the effects of tool flank
wear are given in a flow chart in Figure 5.11.
142
START
Input Tool insert type, work material
Input: Feed rate f, depth of cut d, Cutting speed V, Normal rake angle γn, Inclination angle i, Approach angle Cs, Wearland width VB
Determine the equivalent cutting edge and Find the equivalent variables:i*, γ*n, C*s
Obtain τ, β, rl, Cce, Cte, Ccw and Ctw from the orthogonal cutting Database
Calculate φ*n, β*n
Calculate:
Chip formation forces: Fxs, Fys and Fzs
Edge forces: Fxe, Fye and Fze
Wearland forces: Fxw Fyw and Fzw
The total forces: Fx, Fy and Fz
Output the predicted forces:
Fx, Fy and Fz
End
YesNo
make a new prediction?
Figure 5.11 Procedures for cutting force prediction for turning operations with tool flank wear.
5.6 Model verification
An investigation is needed to qualitatively and quantitatively verify the proposed
cutting force models for turning operations. The plausibility and generality of the
predictive models for turning operations will be assessed by comparing the predicted
143
trends of the cutting forces. To assess the predictive capability of the models, the
cutting forces from the predictive models with the input cutting variables and basic
cutting quantity database will be compared with cutting forces obtained from the
cutting tests under corresponding conditions.
5.6.1 Experimental lay-out and procedures Cutting tests were conducted in order to verify the cutting force model in turning
operations. The measured output variables are the three cutting force components,
i.e. the tangential force Fzm, the feed force Fxm, and the radial force Fym. The cutting
conditions and tool geometry used included cutting speed V, feed rate f, depth of cut
d, tool approach angle Cs, tool normal rake angle γn, inclination angle i, nose radius r,
wearland size VB, work material and tool material.
The tool material and work material used were the same as those used in developing
the basic cutting quantity database so that the force predictions could be carried out.
Work material is CS1020 carbon steel bar of 100mm in diameter and its chemical
compositions were 0.2%C, 0.6%Mn, 0.06%P and 0.06%S, with the tensile strength
of 380 MPa and hardness of 130 BHN. Type TP20 inserts with a nose radius of
0.8mm and flat rake face were used. The inserts were coated with 8µm TiN. Both
‘sharp’ inserts and those with an artificially made wearland were used. The wearland
size of 0.2, 0.4 and 0.6mm were formed by a lapping process.
The approach angle Cs controls the feed force and radial force. The proportion of the
nose radius edge engaged in cutting was affected by the approach angle. For this
reason, two levels of the approach angles (0 degree and 45 degree) were selected in
the cutting tests. When zero degree approach angle was used, three levels of
inclination angles, -5, 0 and 5 degrees, were selected. For a 45 degree approach
angle, the inclination angles used were -5 and 5 degrees. For each of the above
combinations, three levels of normal rake angles were used, i.e. -5, 0 and 5 degrees.
The cutting parameters were selected so that they fell within the boundary of the
parameters used in establishing the cutting quantity database. The cutting speed was
144
selected to avoid built-up edges and within the range of stable operations (without
severe vibrations). Two levels of cutting speeds were selected, namely, 100 and 200
m/min for zero approach tools, 200 and 300m/min for a 45 degree approach angle.
Because the tool nose radius is 0.8mm, the maximum feed rate that may be used is
0.64 mm/revolution (0.8r mm/rev) according to the ISO standard [70].
Considerations were also given to the available feeds on the machine tool and the
capability of the machine tool. Thus, three levels of feed rates were selected; they
were of 0.1, 0.17 and 0.24 mm/revolutions. The above conditions were tested under
three levels of depth of cut for each approach angle, i.e. 1.0, 2.0, and 3.0mm for a
zero degree approach angle, and 0.5, 1.0 and 1.5mm for 45 degree approach angle.
Table 5.1 summaries the above experimental design. Using a full factorial design, a
total of 1080 cutting tests were carried out.
Table 5.1 Cutting parameters for turning operation tests.
Approach angle Cs Parameters Values
Wearland size VB (mm) 0, 0.2, 0.4, 0.6
Normal rake angle γn (deg.) -5o, 0o, 5o
Feed rate f (mm/rev) 0.1, 0.17, 0.24 0o and 45o
Tool nose radius r (mm) 0.8
Inclination angle i (deg.) -5o, 0o ,5o
Cutting speed V (m/min) 100, 200 0o
Depth of cut d (mm) 1, 2, 3
Inclination angle i (degree) -5o, 5o
Cutting speed V (m/min) 200, 300 45o
Depth of cut d (mm) 0.5, 1, 1.5
A Leadwell CNC lathe was used in the tests. After the work bar was mounted
between the centres of the lathe, a preparatory cut was taken to cut the bar to the
diameter of 100mm. (The preparatory cut also removes the surface scale and the
decarburized surface layer and produces an outside diameter concentric with the axis
of the rotation of the lathe.)
145
The cutting forces were measured by a Kistler type 9257A dynamometer at 20
samples per second. After the signals were amplified by three Kistler type 5001
charge amplifiers, they were recorded into a computer using an A/D converter and an
in-house developed data acquisition software program. The final force data for each
cut were taken as the average of 40 samples in the steady cutting stage. The
experimental data is listed in Appendix C.
5.6.2 Results and discussion The generality and plausibility of the cutting force models in turning operations were
investigated by examining the predicted trends with respect to the process
parameters. Using the foregoing equations and procedures given in this chapter, the
prediction of the cutting forces was carried out and compared with the experimental
results based on the cutting conditions described in the previous section. The
tangential force Fz, feed force Fx and radial force Fy are represented in Figures 9-14,
where the lines represent the predicted forces while the symbols represent the
measured forces from the experiments.
0
500
1000
1500
2000
2500
0 0.1 0.2 0.3Feed f (mm/rev)
F (N
)
Fzm (N)Fxm (N)Fym (N)Predicted
V=200m/mind=2mmi=-5deg.γn=5degCs=0deg.VB=0.4mm
Figure 5.12 Predicted and measured forces vs feed rate f.
The Figure 5.12 shows the typical results of the predicted and measured cutting
forces against feed rate f. When the feed increases, the tangential force Fz and feed
force Fx increase significantly. The radial force Fy only shows a little increase due to
the zero approach angle used. These trends are reasonable and have been correctly
146
predicted by the models. The predicted and experimental values are in excellent
agreement both in terms of their trends and the quantitative values.
The typical force trend against the depth of cut is shown in Figure 5.13. The Fz and
Fx increase considerably with an increase in the depth of cut d; but the Fy component
shows a slight decreasing trend with an increase the depth of cut. This is due to the
successful prediction of the equivalent cutting edge and its approach angle that
decreases with the increase of the depth of cut. It is known that the approach angle
mainly controls the feed force Fx and radial force Fy components, so that an increase
in the depth of cut results in a decrease in the equivalent approach angle which in
turn results in a decrease in the radial force component. The predicted forces (both
trends and values) agree very well with the experimental results as shown in Figure
5.13.
An increase in normal rake angle results in a decrease in the predicted and
experimental forces as shown in Figure 5.14, as might be expected for all machining
operations. The predicted forces show good agreement with experimental results.
The cutting forces against the inclinational angle are plotted in Figure 5.15. It can be
seen that the forces decrease slightly with an increase in the inclinational angle. With
a single straight cutting edge such as in classic oblique cutting, varying the cutting
edge inclination angle does not cause a notable change in the cutting force
component, but redistributes the other two force components.
0
500
1000
1500
2000
2500
0 1 2 3 4Depth of cut d (mm)
F (N
)
Fzm (N)Fxm (N)Fym (N)Predicted
V=200m/minf=0.17mm/ri=5deg.γn=5degCs=0degVB=0.6mm
Figure 5.13 Predicted and measured forces vs depth of cut d.
147
0
500
1000
1500
2000
2500
-10 -5 0 5 10Normal rake angle γn (degree)
F (N
)
FzmFxmFymPredicted
V=200m/min f=0.24mm/ri=0degCs=0degd=2mmVB=0.4mm
Figure 5.14 Predicted and measured forces vs normal rake angle γγγγn.
When cutting with a nose radius edge, the inclination angle defined on the major
cutting edge affects the magnitude of the rake angle as well as the inclination angle
on the nose radius edge, i.e. an increase in the inclination angle on the major cutting
edge increases the rake angle on the nose radius edge, which in turn increases the
normal rake angle on the equivalent cutting edge. As a result, the cutting forces are
reduced. The cutting force predictive models have correctly predicted this trend and
the predicted force values are in good agreement with experimental results.
0
500
1000
1500
2000
2500
-10 -5 0 5 10Inclination angle i (degree)
F (N
)
FzmFxmFymPredicted
V=200m/min f=0.1mm/rγn=5degCs=0degd=3mmVB=0.2mm
Figure 5.15 Predicted and measured forces vs inclination angle i.
148
The cutting forces against the wearland size are shown in Figure 5.16. An increase in
the wearland size results in an increase in all the three force components. This may
be expected as an increase in the wearland size increases the rubbing forces on the
wearland and increases the cutting forces. The models have correctly predicted this
trend. Quantitatively, the models’ predictions are in good agreement with the
corresponding experimental data.
0
500
1000
1500
2000
2500
0 0.2 0.4 0.6 0.8Wearland VB (mm)
F (N
)
FzmFxmFymPredicted
V=200m/min f=0.1mm/rγn=0degCs=0degi=0degd=3mm
Figure 5.16 Predicted and measured forces vs wearland size VB.
The influence of the approach angle on the cutting forces is plotted in Figure 5.17.
The sold lines represent the predicted forces for the approach angle of 45 degrees,
and the dash lines represent the predicted forces for approach angle at zero degree.
The tangential force Fz shows a slight increase with the increase of the approach
angle in Figure 5.17 (a); the radial force Fy shows a considerable increase with the
increase of the approach angle show in Figure 5.17 (b); while the increase in the feed
force Fx is very marginal in Figure 5.17 (c). It can be seen that the models have
correctly predicted these trends.
149
0
500
1000
1500
2000
2500
0 0.1 0.2 0.3Feed f (mm/rev)
Fz (N
)
Cs=45degCs=0deg
V=200m/min γn=-5degi=5degd=1mmVB=0.4mm
Predicted Cs=45degPredicted Cs=0deg
(a)
0
500
1000
1500
2000
2500
0 0.1 0.2 0.3Feed f (mm/rev)
Fy (N
)
Cs=45degCs=0deg
V=200m/min γn=-5degi=5degd=1mmVB=0.4mm
Predicted Cs=45degPredicted Cs=0deg
(b)
0
500
1000
1500
2000
2500
0 0.1 0.2 0.3Feed f (mm/rev)
Fx (N
)
Cs=45degCs=0deg
V=200m/min γn=-5degi=5degd=1mmVB=0.4mm
Predicted Cs=45degPredicted Cs=0deg
(c)
Figure 5.17 Predicted and measured forces vs feed rate for different approach angle Cs.
150
-45 -30 -15 0 15 30 45Percentage deviation (Fz)
0%
10%
20%
30%
40%
50%
Freq
uenc
y %
Mean=8.1%Std.Dev.=13.55%
(a)
-45 -30 -15 0 15 30 45
Percentage deviation (Fy)
0%
10%
20%
30%
40%
50%
Freq
uenc
y %
Mean=2.2%Std.Dev=32.1%
(b)
-45 -30 -15 0 15 30 45Percentage deviation (Fx)
0%
10%
20%
30%
40%
50%
Freq
uenc
y %
Mean=9.5%Std.Dev=22.38
(c)
Figure 5.18 Percentage deviations of predicted forces from the experimental forces.
151
The above analysis has confirmed that the cutting force models have been formulated
in the correct form as they can give reasonable prediction of the trends of the cutting
forces with respect to the process variables. In order to assess the adequacy or
predictive capability of the models, a quantitative evaluation is needed.
The quantitative comparison was made based on the percentage deviation of the
predicted cutting force component from the corresponding experimental data. The
results are given in the histograms in Figure 5.18. The figure shows that the models’
prediction of the tangential force, Fz, yields an average percentage deviation of 8.1%
with a standard deviation of 13.55%, as shown in Figure 5.18(a). The average
percentage deviation for the feed force, Fx, is 2.2% with a 32.1% standard deviation,
as shown in Figure 5.18(b). For the radial force Fy, the average percentage deviation
is 9.5% with a standard deviation of 22.38%, as shown in Figure 5.18(c). The
variations of standard deviation are mainly caused by experimental errors.
Thus, the cutting force models developed in this chapter can give adequate
predictions for the range of the experimental conditions used in this study. It also
shows that only the basic cutting quantity database from the orthogonal cutting tests
is needed to predict the cutting forces in practical operations. The successful
prediction of the cutting forces has confirmed the models’ abilities to allow for the
effects of flank wear in turning operations.
5.7 Concluding remarks The cutting force models for turning operations have been developed so that they
take into account the effect of tool flank wear. The models were an extension of the
mechanics of cutting analysis for the oblique cutting with tool wear in the previous
chapter. The equivalent cutting edge concept was used in the modelling process to
allow for the effects of tool nose radius on practical cutting tools. The turning
operation has then been related to an oblique cutting process with reference to an
equivalent cutting edge. The forces required for chip formation were modelled by
using the thin shear zone analysis with respect to the equivalent cutting edge and its
equivalent geometrical angles. The basic cutting quantity database developed in the
152
orthogonal cutting has been used in the shear zone analysis to develop the predictive
models of the cutting force for chip formation in turning operations.
An analysis of the edge forces has been carried out for turning operations with nose
radius tools. The edge force models have subsequently been developed. Similarly,
the wearland force models have been developed for tool with nose radius in turning
operations. Based on the analysis of the oblique cutting in the previous chapter, the
overall cutting force models have been proposed which consider the three sources of
forces, i.e. the force for basic chip formation, the edge force and the wearland force.
A wide range of cutting parameters was selected in the cutting tests to verify the
predictive cutting force models proposed for turning operations allowing for the
effects of tool flank wear. The three cutting force components were measured for
each cut in the experiments and used to compare with the predicted forces from the
cutting force models under the corresponding cutting conditions.
The analysis of the trends of the predicted forces was carried out by plotting the
forces against the cutting variables. The measured forces were also plotted in the
figures to assess the predictive ability by comparing the trends of the predicted and
experimental forces. It has been shown that the trends of the predicted forces agree
very well with the corresponding experimental results.
The quantitative assessment of the predictive cutting force models has been
conducted by the statistical analysis of the percentage deviation for the predicted
forces with respect to experiment results. The histograms of the percentage deviation
of the predicted forces from the corresponding experimental data were used to assist
this analysis. The average and standard deviations in the histograms showed that the
developed cutting force models for turning operations can give excellent predictions
under the cutting conditions used in this study.
Thus, predictive cutting force models allowing for the effects of tool flank wear have
been verified to be able to adequately predict the cutting forces in turning operations
when tool flank wear is present. These models have provided a unique means of
predicting the cutting forces as tool wear progresses for real time adaptive control as
153
well as to sense tool wear and imminent tool failure in tool condition monitoring in
turning operations.
This study has also shown that only the basic cutting quantities need to be
determined from orthogonal cutting tests in order to use this mechanics of cutting
approach to predict the cutting forces. The predictive force models developed in this
study can be considered as generic for turning steel with continuous chip formation.
When the tool and work materials are changed, only the basic cutting quantities such
as the shear angle and share stress need to be determined before applying these
models.
The modelling approach using the mechanics of cutting analysis developed in this
project for turning operations with tool flank wear has the potential for usage in any
other machining operations, such as milling and drilling, to develop the predictive
cutting force models allowing for the effects of tool flank wear.
154
Chapter 6 Final conclusions and future work
6.1 Final conclusions
An extensive literature review has been carried out at beginning of the thesis; the
emphasis was placed on the fundamental studies and analysis of the classical
orthogonal and oblique cutting processes. The mechanics of cutting analysis in
orthogonal and oblique cuttings and associated turning operations was reviewed first,
followed by an analysis of edge force and the introduction of this force into cutting
force models. The effect of the tool wear, in particular flank wear, on the cutting
forces was discussed based on the literature published. It has been shown that there is
a general lack of fundamental study and understanding of how the tool flank wear
affects the chip formation process in the shear zone and the cutting process as a
whole, as well as the approaches and models to predict the cutting forces when tool
flank wear is present.
Although the orthogonal cutting models and subsequently the oblique cutting models
have been established and further developed for more than half a century, early work
was limited to relative simple classical orthogonal cutting and more general oblique
cutting processes. The edge force was ignored in early models. This caused
inaccuracies in the evaluation of the friction angle and shear stress as well as the
cutting forces. The edge force was introduced in the mechanics of cutting analysis in
the early 1980s and the predictive cutting force models were developed to allow for
the effect of the edge force. The nose radius in a turning operation was also
considered and it was found that it could not be ignored in the analysis. Both
‘equivalent cutting edge’ and ‘generalised mechanics of cutting analysis’ were all
used to allow for the effect of the nose radius in the turning operation, but the
equivalent cutting edge is more favourable in practice. The review has been unable to
reveal any notable work to study the effect of tool flank wear on the chip formation
process and the overall cutting process in any machining operations.
A comprehensive experimental investigation was carried out on the orthogonal
cutting process in this thesis. The study has found that when tool flank wear is
155
present, the basic cutting quantities (shear angle, friction angle, shear stress and shear
angle relationship) have shown similar trends to those with ‘sharp’ tools. As a
significant finding of this research, it has been found for the first time that tool flank
wear does not qualitatively and quantitatively affect the basic chip formation or basic
cutting process in orthogonal cutting. The increase in the cutting force is a result of
the rubbing force between the tool flank and the work surface. The practical
implication of this finding is that it is possible to model the machining processes by
using the well developed machining theories for ‘sharp’ tools with the addition of the
rubbing forces on the wearland, rather than spending time developing new machining
theories for worn tools, which may be contrary to the well-recognized models.
To gain a full understanding of the wearland forces and for modelling the cutting
forces in machining operations, the rubbing forces on the flank have been fully
analysed and empirical models developed. Similarly, the edge forces have been
separated from the total cutting forces and an empirical model has been established
for use in modelling the cutting forces late in this thesis. As the first step to model
the cutting forces using the proposed new approach, i.e. using the existing machining
theories and taking into account the werland forces, the cutting force models for
orthogonal cutting allowing for the effect of tool flank wear have been developed. A
numerical study has found an excellent agreement between the predicted and
experimental forces.
Based on the mechanics of cutting analysis, the predictive force models for oblique
cutting allowing for tool flank wear were then developed. From the orthogonal
cutting analysis, the total force consists of three components, the forces required to
perform the chip formation in the shear zone, the edge forces on the cutting edge, and
the forces on the wearland (or the wearland forces). The edge force obtained in
orthogonal cutting analysis was reconsidered in the oblique cutting according to a
geometrical and mechanics of cutting analysis. Similarly, the wearland force model
developed in the orthogonal cutting analysis has been applied to oblique cutting
through a mechanics of cutting analysis. Finally, the basic cutting quantity database
developed in the orthogonal cutting study was used when evaluating the forces
required when shearing the workpiece using the thin shear zone model. The overall
cutting forces models for oblique cutting with tool wear have been qualitatively and
156
quantitatively assessed by an experiment. An accurate prediction of the cutting forces
has been achieved, confirming the correctness of the findings in Chapter 3 and the
new modelling approach for machining with tool flank wear.
The mechanics of cutting analysis has been extended and applied to turning
operations to predict the cutting forces allowing for the effect of tool flank wear. The
equivalent cutting edge concept was used to take into account the effect of tool nose
radius. The equivalent tool geometrical angles with respect to the equivalent cutting
edge were then been determined. As a result, the oblique cutting analysis and models
developed in Chapter 4 for the forces required to shear the material in the shear zone
have been applied to the equivalent cutting edge to evaluate this part of the cutting
forces. An analysis was then made to apply the edge force and wearland force
models to turning with a nose radius. A wide range of cutting parameters has been
selected to verify the cutting force models in turning operations. The trends of the
predicted forces agree well with the experimental results under corresponding cutting
conditions. The histograms for the percentage deviations of the predicted forces from
the corresponding experimental data have shown excellent prediction results.
6.2 Major Scientific Contributions of the Research This thesis has found for the first time that tool flank wear does not affect the basic
cutting or chip formation process in the shear zone, and it is the rubbing or ploughing
force on the wearland that causes the increase of the overall cutting forces. Based on
this finding, a new modelling approach has been proposed which makes full use of
existing machining theories, taking into consideration the additional rubbing force on
the wearland. The modelling approach has been verified to be correct and viable
when applied to classical orthogonal and oblique cutting and turning operations. This
study opens new ways to model and accurately predict the cutting forces in practical
machining operations effective and efficient use these processes for further economic
gains. Together with the cutting forces model which was developed in this thesis,
these models have applications in solving cutting force prediction in tool wear
cutting conditions.
157
6.3 Proposals for further studies From the investigations conducted in this thesis, a number of possible avenues for
future work can be suggested.
More tool geometries and tool-work material combinations may be tested to make
sure that the modelling approach developed in this thesis is generic. A
comprehensive orthogonal cutting test would require a wide range of tool
geometrical features and cutting conditions to establish the basic cutting quantity
database for different tool and work materials.
The wearland force model used in the thesis was obtained from the experimental data
by a statistical analysis. A more fundamental approach is suggested to model the
rubbing or ploughing process and develop the rubbing force model.
The models developed in this thesis can be extended to other machining operations,
such as milling and drilling. The modelling approach could also be used in high
speed machining operations. Such investigations would provide further evidence of
the applicability and generality of the modelling approach developed in this thesis.
158
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A1
Appendix A: Measured forces, Chip length and weight in
orthogonal cutting
V (m/min)
t (mm/r)
γn (deg.)
VB (mm)
Lc (mm)
Wc (g)
Fcm (N)
Ftm (N)
100 0.1 0 0 430 3.479 815.0 542.5 100 0.17 0 0 460 5.763 1201.2 755.5 100 0.24 0 0 502 9.544 1638.5 971.1 100 0.31 0 0 527 11.65 2092.7 1412.1 150 0.1 0 0 575 3.932 797.6 567.7 150 0.17 0 0 400 4.783 1209.5 796.2 150 0.24 0 0 490 8.207 1635.6 1036.7 150 0.31 0 0 444 9.135 2073.2 1242.1 200 0.1 0 0 553 3.912 802.9 610.9 200 0.17 0 0 440 4.955 1185.5 758.7 200 0.24 0 0 490 7.869 1586.6 948.5 200 0.31 0 0 433 8.512 1958.0 1105.7 100 0.1 0 0.2 445 3.378 887.1 655.1 100 0.17 0 0.2 415 5.22 1287.6 875.3 100 0.24 0 0.2 506 9.888 1777.9 1097.7 100 0.31 0 0.2 457 10.717 2144.0 1391.6 150 0.1 0 0.2 480 3.47 874.5 656.6 150 0.17 0 0.2 435 5.3 1340.2 937.7 150 0.24 0 0.2 540 8.762 1734.0 1128.3 150 0.31 0 0.2 452 10.479 2281.1 1470.0 200 0.1 0 0.2 613 4.237 870.4 562.2 200 0.17 0 0.2 345 4.015 1255.9 799.4 200 0.24 0 0.2 505 6.863 1651.1 973.7 200 0.31 0 0.2 407 8.15 2060.0 1160.7 100 0.1 0 0.4 532 4.056 871.1 536.2 100 0.17 0 0.4 435 5.304 1288.0 883.2 100 0.24 0 0.4 362 6.229 1864.7 1313.2 150 0.1 0 0.4 540 3.942 891.7 667.4 150 0.17 0 0.4 330 4.172 1361.2 939.7 150 0.24 0 0.4 391 6.695 1728.0 1113.6 150 0.31 0 0.4 200 4.259 2128.5 1318.0 200 0.1 0 0.4 605 4.179 1177.8 1150.2 200 0.17 0 0.4 414 4.466 1646.5 1379.5 200 0.24 0 0.4 498 7.57 2012.5 1469.8 100 0.1 0 0.6 354.9 2.19 1138.3 893.6 100 0.17 0 0.6 287 3.37 1558.2 1101.4 100 0.24 0 0.6 204 3.042 1978.9 1458.1 100 0.31 0 0.6 183 4.16 2406.9 1645.2 150 0.1 0 0.6 402 2.928 1079.9 802.0 150 0.17 0 0.6 1164 14.087 1471.1 1007.9 150 0.24 0 0.6 486.5 7.792 1897.8 1243.6 150 0.31 0 0.6 354 7.765 2164.2 1391.7
A2
200 0.1 0 0.6 580 4.509 1034.3 793.7 200 0.17 0 0.6 481.5 5.553 1427.8 971.2 200 0.24 0 0.6 625.5 10.377 1788.2 1125.0 200 0.31 0 0.6 421.5 9.575 2333.2 1455.5 100 0.1 5 0 382.5 3.154 809.1 535.9 100 0.17 5 0 472.5 5.78 1169.0 672.0 100 0.24 5 0 517 9.056 1540.9 817.6 100 0.31 5 0 192.5 4.519 1965.8 1201.8 150 0.1 5 0 412.5 2.774 746.6 488.2 150 0.17 5 0 495 5.899 1126.8 500.9 150 0.24 5 0 757 11.665 1548.1 940.6 150 0.31 5 0 444 8.676 1908.9 1086.1 200 0.1 5 0 1005.5 6.799 755.6 501.2 200 0.17 5 0 533 6.023 1160.9 732.0 200 0.24 5 0 690.5 10.375 1505.1 858.2 200 0.31 5 0 345.5 10.035 1860.8 969.4 100 0.1 5 0.2 370 2.628 799.7 518.3 100 0.17 5 0.2 530 6.368 1250.8 771.2 100 0.24 5 0.2 530 9.572 1721.7 958.3 100 0.31 5 0.2 140 3.38 1910.7 1057.9 150 0.1 5 0.2 775 5.34 818.4 548.6 150 0.17 5 0.2 490 5.419 1213.0 752.1 150 0.24 5 0.2 613 9.788 1633.2 979.4 150 0.31 5 0.2 475 9.189 1932.4 1117.2 200 0.1 5 0.2 453 3.214 840.0 573.0 200 0.17 5 0.2 590 6.512 1213.7 764.9 200 0.24 5 0.2 685 10.043 1546.8 874.5 200 0.31 5 0.2 435 8.315 1885.5 1018.7 100 0.1 5 0.4 457 3.272 848.2 615.4 100 0.17 5 0.4 486 5.648 1244.1 810.0 100 0.24 5 0.4 543.5 9.41 1682.7 978.8 100 0.31 5 0.4 323 6.481 1847.3 1033.5 150 0.1 5 0.4 702 4.787 906.9 626.8 150 0.17 5 0.4 531 6.153 1287.3 799.2 150 0.24 5 0.4 593.5 9.199 1632.8 952.2 150 0.31 5 0.4 415.5 8.691 2022.9 1119.0 200 0.1 5 0.4 558.5 4.055 912.7 609.1 200 0.17 5 0.4 460 4.97 1202.7 749.6 200 0.24 5 0.4 447.5 6.591 1558.4 888.9 200 0.31 5 0.4 666.5 12.8 1886.8 982.7 100 0.1 5 0.6 430 2.887 883.7 641.5 100 0.17 5 0.6 553 6.148 1316.2 888.8 100 0.24 5 0.6 320 4.897 1727.5 1034.7 100 0.31 5 0.6 335 7.806 1965.1 1235.2 150 0.1 5 0.6 436 2.929 990.8 710.8 150 0.17 5 0.6 464 5.293 1351.3 852.2 150 0.24 5 0.6 706.5 10.965 1692.9 999.9 150 0.31 5 0.6 552 11.761 2772.8 1598.8 200 0.1 5 0.6 521 3.577 908.6 578.8
A3
200 0.17 5 0.6 541.5 5.976 1226.6 700.1 200 0.24 5 0.6 605.5 8.84 1900.9 946.4 200 0.31 5 0.6 718 13.399 1552.2 823.5 100 0.1 -5 0 380 3.579 947.4 734.8 100 0.17 -5 0 505 7.441 1426.9 1089.3 100 0.24 -5 0 402 7.255 1707.2 1285.1 100 0.31 -5 0 410 9.171 2161.3 1604.6 150 0.1 -5 0 290 2.21 879.1 701.6 150 0.17 -5 0 338 4.486 1373.5 1060.9 150 0.24 -5 0 447 7.717 1748.9 1268.0 150 0.31 -5 0 545 11.839 2148.6 1561.3 200 0.1 -5 0 481 3.629 872.5 733.7 200 0.17 -5 0 476 5.636 1256.7 957.8 200 0.24 -5 0 662 10.807 1627.9 1136.4 200 0.31 -5 0 462 10.173 2111.9 1399.2 100 0.1 -5 0.2 457 5.309 924.9 784.0 100 0.17 -5 0.2 523 7.906 1380.6 1105.8 100 0.24 -5 0.2 340 6.206 1661.7 1236.2 100 0.31 -5 0.2 456 10.603 2114.7 1571.1 150 0.1 -5 0.2 395 3.323 887.5 763.4 150 0.17 -5 0.2 514 6.86 1372.5 1093.1 150 0.24 -5 0.2 451 8.126 1723.9 1273.9 150 0.31 -5 0.2 501 11.013 2034.0 1393.6 200 0.1 -5 0.2 484 3.97 865.3 722.1 200 0.17 -5 0.2 475 6.3 1273.2 931.3 200 0.24 -5 0.2 492 8.518 1669.0 1139.0 200 0.31 -5 0.2 515 11.443 2042.2 1318.6 100 0.1 -5 0.4 434 3.487 1172.8 898.1 100 0.17 -5 0.4 125 1.587 1698.2 1306.6 100 0.24 -5 0.4 256 4.489 2018.4 1507.0 100 0.31 -5 0.4 467 10.422 2369.8 1731.4 150 0.1 -5 0.4 442 3.311 1185.2 964.6 150 0.17 -5 0.4 460 5.837 1621.3 1232.5 150 0.24 -5 0.4 473 8.619 2104.9 1564.0 150 0.31 -5 0.4 284 6.033 2348.9 1576.3 200 0.1 -5 0.4 525 3.814 1137.4 919.7 200 0.17 -5 0.4 507 5.712 1546.8 1164.7 200 0.24 -5 0.4 414 7.043 1977.7 1377.5 200 0.31 -5 0.4 408 8.263 2235.3 1365.2 100 0.1 -5 0.6 542 4.04 1451.8 1231.3 100 0.17 -5 0.6 150 1.456 1517.5 1182.3 100 0.24 -5 0.6 440 7.471 1985.3 1479.8 100 0.31 -5 0.6 464 10.245 2451.8 1781.1 150 0.1 -5 0.6 494 4.393 1141.9 968.9 150 0.17 -5 0.6 650 7.452 1571.9 1231.2 150 0.24 -5 0.6 415 7.605 2037.4 1545.1 150 0.31 -5 0.6 387 8.253 2397.2 1658.2 200 0.1 -5 0.6 425 3.32 1100.6 839.1 200 0.17 -5 0.6 405 4.778 1520.7 1066.9
A4
200 0.24 -5 0.6 519 8.804 1913.5 1259.5 200 0.31 -5 0.6 499 10.256 2224.8 1408.8
A5
Appendix B: Measured forces in oblique cutting
γn
(deg.) i
(deg.) V
(m/min) VB
(mm) t
(mm/r) Fcm(N) Ftm(N) Frm(N)
0 -5 100 0.2 0.1 912.8 694.5 -20.4 0 -5 100 0.2 0.17 1300.0 903.4 -32.6 0 -5 100 0.2 0.24 1850.7 1309.3 -57.4 0 -5 200 0.2 0.1 882.9 677.4 -29.2 0 -5 200 0.2 0.17 1249.9 857.5 -42.7 0 -5 200 0.2 0.24 1661.5 1036.8 -73.9 0 -5 100 0.4 0.1 1013.5 841.4 -23.4 0 -5 100 0.4 0.17 1472.9 1112.6 -43.9 0 -5 100 0.4 0.24 1966.9 1375.6 -88.1 0 -5 200 0.4 0.1 965.9 699.2 -34.4 0 -5 200 0.4 0.17 1355.7 879.8 -62.8 0 -5 200 0.4 0.24 1695.3 1034.1 -78.8 0 -5 100 0.6 0.1 1085.2 893.2 -42.8 0 -5 100 0.6 0.17 1539.5 1117.6 -59.3 0 -5 100 0.6 0.24 1986.7 1474.3 -69.0 0 -5 200 0.6 0.1 984.7 779.0 -32.1 0 -5 200 0.6 0.17 1371.9 953.9 -49.7 0 -5 200 0.6 0.24 1703.8 1042.9 -71.8 5 -5 100 0.2 0.1 854.7 612.3 -10.9 5 -5 100 0.2 0.17 1196.6 748.7 -23.8 5 -5 100 0.2 0.24 1571.1 950.6 -33.7 5 -5 200 0.2 0.1 807.0 569.4 -12.4 5 -5 200 0.2 0.17 1188.9 735.9 -28.2 5 -5 200 0.2 0.24 1502.3 846.7 -45.4 5 -5 100 0.4 0.1 982.1 783.4 0.7 5 -5 100 0.4 0.17 1328.7 897.8 -24.2 5 -5 100 0.4 0.24 1676.5 1053.3 -39.0 5 -5 200 0.4 0.1 870.1 587.8 -17.8 5 -5 200 0.4 0.17 1187.9 733.2 -29.9 5 -5 200 0.4 0.24 1522.1 872.0 -49.8 5 -5 100 0.6 0.1 1060.3 845.3 -9.0 5 -5 100 0.6 0.17 1402.2 982.8 -18.2 5 -5 100 0.6 0.24 1850.5 1282.9 -31.8 5 -5 200 0.6 0.1 920.2 665.1 -15.7 5 -5 200 0.6 0.17 1260.4 800.5 -31.4 5 -5 200 0.6 0.24 1627.7 943.3 -50.1 -5 -5 100 0.2 0.1 955.1 765.7 -29.0 -5 -5 100 0.2 0.17 1384.7 1038.6 -52.0 -5 -5 100 0.2 0.24 1929.4 1467.9 -80.8 -5 -5 200 0.2 0.1 915.7 734.5 -27.2 -5 -5 200 0.2 0.17 1329.6 951.4 -55.7 -5 -5 200 0.2 0.24 1733.9 1140.8 -74.0 -5 -5 100 0.4 0.1 1027.8 836.2 -18.8 -5 -5 100 0.4 0.17 1461.5 1101.6 -41.5 -5 -5 100 0.4 0.24 1974.3 1477.7 -62.1 -5 -5 200 0.4 0.1 976.3 784.0 -24.7 -5 -5 200 0.4 0.17 1390.3 1000.5 -49.8
A6
-5 -5 200 0.4 0.24 1797.3 1187.0 -75.4 -5 -5 100 0.6 0.1 1231.3 1059.5 -41.2 -5 -5 100 0.6 0.17 1687.9 1341.3 -78.5 -5 -5 100 0.6 0.24 2074.8 1568.8 -105.0 -5 -5 200 0.6 0.1 1069.7 836.0 -48.9 -5 -5 200 0.6 0.17 1490.2 1023.3 -80.3 -5 -5 200 0.6 0.24 1852.7 1217.6 -102.6 -5 5 100 0.2 0.1 957.0 774.3 45.7 -5 5 100 0.2 0.17 1377.2 1024.6 72.7 -5 5 100 0.2 0.24 1914.5 1470.6 90.9 -5 5 200 0.2 0.1 921.2 751.5 45.6 -5 5 200 0.2 0.17 1331.0 958.7 73.8 -5 5 200 0.2 0.24 1742.4 1158.8 96.7 -5 5 100 0.4 0.1 1011.8 844.1 48.6 -5 5 100 0.4 0.17 1453.9 1141.6 80.3 -5 5 100 0.4 0.24 1965.2 1533.4 104.7 -5 5 200 0.4 0.1 945.6 778.6 43.7 -5 5 200 0.4 0.17 1353.1 1000.2 66.5 -5 5 200 0.4 0.24 1756.5 1169.8 92.0 -5 5 100 0.6 0.1 1251.1 1043.4 59.2 -5 5 100 0.6 0.17 1669.8 1308.0 81.2 -5 5 100 0.6 0.24 2110.7 1595.6 86.7 -5 5 200 0.6 0.1 1076.1 847.6 47.3 -5 5 200 0.6 0.17 1483.2 1050.2 74.9 -5 5 200 0.6 0.24 1861.7 1232.1 97.2 0 5 100 0.2 0.1 845.5 656.0 26.2 0 5 100 0.2 0.17 1249.1 847.6 25.7 0 5 100 0.2 0.24 1632.6 1048.8 41.9 0 5 200 0.2 0.1 796.3 609.9 24.2 0 5 200 0.2 0.17 1184.0 817.9 37.3 0 5 200 0.2 0.24 1576.5 985.9 49.5 0 5 100 0.4 0.1 864.8 652.9 23.1 0 5 100 0.4 0.17 1263.6 879.8 39.8 0 5 100 0.4 0.24 1693.2 1172.4 50.7 0 5 200 0.4 0.1 873.9 684.7 22.8 0 5 200 0.4 0.17 1240.4 848.0 39.2 0 5 200 0.4 0.24 1609.3 1013.6 49.7 0 5 100 0.6 0.1 928.5 738.1 22.7 0 5 100 0.6 0.17 1319.1 984.1 36.9 0 5 100 0.6 0.24 1806.6 1352.0 45.6 0 5 200 0.6 0.1 859.8 687.7 26.5 0 5 200 0.6 0.17 1253.5 890.9 43.0 0 5 200 0.6 0.24 1634.1 1054.5 46.0 5 5 100 0.2 0.1 774.7 529.0 35.5 5 5 100 0.2 0.17 1124.6 690.9 53.1 5 5 100 0.2 0.24 1507.1 893.9 66.6 5 5 200 0.2 0.1 795.8 573.4 35.7 5 5 200 0.2 0.17 1156.6 733.0 56.8 5 5 200 0.2 0.24 1493.2 871.6 71.9 5 5 100 0.4 0.1 886.0 671.4 42.5 5 5 100 0.4 0.17 1239.9 815.7 52.5
A7
5 5 100 0.4 0.24 1599.0 970.1 67.6 5 5 200 0.4 0.1 825.8 595.7 37.8 5 5 200 0.4 0.17 1177.5 747.8 54.8 5 5 200 0.4 0.24 1514.0 883.5 67.1 5 5 100 0.6 0.1 936.7 748.2 30.7 5 5 100 0.6 0.17 1280.4 892.5 49.8 5 5 100 0.6 0.24 1640.6 1025.8 62.2 5 5 200 0.6 0.1 853.1 622.1 40.5 5 5 200 0.6 0.17 1204.1 776.8 52.7 5 5 200 0.6 0.24 1536.7 896.2 67.8
A8
Appendix C: Measured forces in turning operation
Ref. Number
Tool Num.
V (min/m)
f (mm/r)
d (mm)
Cs (deg.)
i (deg)
γn (deg)
VB (mm)
Fxm (N)
Fym (N)
Fzm (N)
#1-00-1 1 100 0.1 1 0 -5 0 0 262.1 173.1 419.0 #1-00-2 1 100 0.1 2 0 -5 0 0 468.4 169.5 713.4 #1-00-3 1 100 0.1 3 0 -5 0 0 697.3 216.7 1066.2 #1-02-1 1 100 0.1 1 0 -5 0 0.2 267.3 147.6 412.3 #1-02-2 1 100 0.1 2 0 -5 0 0.2 523.5 155.8 767.0 #1-02-3 1 100 0.1 3 0 -5 0 0.2 781.0 199.9 1145.1 #1-04-1 1 100 0.1 1 0 -5 0 0.4 306.0 166.1 469.0 #1-04-2 1 100 0.1 2 0 -5 0 0.4 562.6 154.5 805.5 #1-04-3 1 100 0.1 3 0 -5 0 0.4 850.7 192.0 1218.5 #1-06-1 1 100 0.1 1 0 -5 0 0.6 320.7 164.0 485.7 #1-06-2 1 100 0.1 2 0 -5 0 0.6 559.3 172.5 820.6 #1-06-3 1 100 0.1 3 0 -5 0 0.6 854.3 197.4 1225.1 #2-00-1 2 100 0.1 1 0 -5 5 0 201.7 149.7 374.4 #2-00-2 2 100 0.1 2 0 -5 5 0 376.4 157.4 675.4 #2-00-3 2 100 0.1 3 0 -5 5 0 565.3 170.1 983.8 #2-02-1 2 100 0.1 1 0 -5 5 0.2 235.2 119.0 378.9 #2-02-2 2 100 0.1 2 0 -5 5 0.2 441.4 117.0 699.6 #2-02-3 2 100 0.1 3 0 -5 5 0.2 653.3 146.7 1044.0 #2-04-1 2 100 0.1 1 0 -5 5 0.4 227.3 128.0 375.4 #2-04-2 2 100 0.1 2 0 -5 5 0.4 446.9 144.8 722.1 #2-04-3 2 100 0.1 3 0 -5 5 0.4 669.4 132.0 1025.5 #2-06-1 2 100 0.1 1 0 -5 5 0.6 328.1 151.9 455.1 #2-06-2 2 100 0.1 2 0 -5 5 0.6 767.3 193.2 893.6 #2-06-3 2 100 0.1 3 0 -5 5 0.6 1111.5 226.1 1322.7 #3-00-1 3 100 0.1 1 0 -5 -5 0 284.3 168.3 421.8 #3-00-2 3 100 0.1 2 0 -5 -5 0 525.5 194.8 766.7 #3-00-3 3 100 0.1 3 0 -5 -5 0 771.5 227.4 1117.4 #3-02-1 3 100 0.1 1 0 -5 -5 0.2 303.1 155.3 427.1 #3-02-2 3 100 0.1 2 0 -5 -5 0.2 522.1 151.2 743.6 #3-02-3 3 100 0.1 3 0 -5 -5 0.2 810.9 217.5 1148.2 #3-04-1 3 100 0.1 1 0 -5 -5 0.4 364.0 177.0 484.0 #3-04-2 3 100 0.1 2 0 -5 -5 0.4 624.0 165.0 836.3 #3-04-3 3 100 0.1 3 0 -5 -5 0.4 967.9 202.3 1258.8 #3-06-1 3 100 0.1 1 0 -5 -5 0.6 389.3 210.5 513.1 #3-06-2 3 100 0.1 2 0 -5 -5 0.6 671.6 205.3 896.8 #3-06-3 3 100 0.1 3 0 -5 -5 0.6 981.5 222.0 1297.9 #4-00-1 4 100 0.1 1 0 5 -5 0 262.5 119.4 398.8 #4-00-2 4 100 0.1 2 0 5 -5 0 485.7 78.4 702.3 #4-00-3 4 100 0.1 3 0 5 -5 0 748.8 112.5 1101.1 #4-02-1 4 100 0.1 1 0 5 -5 0.2 305.4 147.6 444.5 #4-02-2 4 100 0.1 2 0 5 -5 0.2 521.3 124.2 756.6 #4-02-3 4 100 0.1 3 0 5 -5 0.2 0.0 0.0 0.0 #4-04-1 4 100 0.1 1 0 5 -5 0.4 323.2 152.3 461.7 #4-04-2 4 100 0.1 2 0 5 -5 0.4 569.1 132.7 800.9 #4-04-3 4 100 0.1 3 0 5 -5 0.4 835.0 125.6 1162.0 #4-06-1 4 100 0.1 1 0 5 -5 0.6 321.4 157.2 462.5 #4-06-2 4 100 0.1 2 0 5 -5 0.6 549.7 118.9 785.3 #4-06-3 4 100 0.1 3 0 5 -5 0.6 813.7 134.4 1156.8 #9-00-1 9 100 0.1 1 0 5 0 0 251.3 132.6 402.3 #9-00-2 9 100 0.1 2 0 5 0 0 457.0 122.1 715.1 #9-00-3 9 100 0.1 3 0 5 0 0 733.2 129.1 1087.2
A9
#9-02-1 9 100 0.1 1 0 5 0 0.2 294.0 173.3 460.5 #9-02-2 9 100 0.1 2 0 5 0 0.2 526.0 161.8 800.1 #9-02-3 9 100 0.1 3 0 5 0 0.2 751.3 167.9 1139.5 #9-04-1 9 100 0.1 1 0 5 0 0.4 350.5 195.2 517.8 #9-04-2 9 100 0.1 2 0 5 0 0.4 607.7 174.6 869.1 #9-04-3 9 100 0.1 3 0 5 0 0.4 855.2 182.3 1206.2 #9-06-1 9 100 0.1 1 0 5 0 0.6 389.7 186.8 536.5 #9-06-2 9 100 0.1 2 0 5 0 0.6 672.9 183.5 938.8 #9-06-3 9 100 0.1 3 0 5 0 0.6 975.6 168.9 1322.4
#10-00-1 10 100 0.1 1 0 5 5 0 217.2 127.2 382.3 #10-00-2 10 100 0.1 2 0 5 5 0 413.6 116.6 698.5 #10-00-3 10 100 0.1 3 0 5 5 0 600.1 111.7 1005.4 #10-02-1 10 100 0.1 1 0 5 5 0.2 290.0 172.2 461.6 #10-02-2 10 100 0.1 2 0 5 5 0.2 491.0 128.3 745.9 #10-02-3 10 100 0.1 3 0 5 5 0.2 721.1 157.9 1119.7 #10-04-1 10 100 0.1 1 0 5 5 0.4 257.9 105.7 399.8 #10-04-2 10 100 0.1 2 0 5 5 0.4 465.1 97.4 731.7 #10-04-3 10 100 0.1 3 0 5 5 0.4 729.2 87.6 1112.0 #10-06-1 10 100 0.1 1 0 5 5 0.6 339.0 186.1 499.4 #10-06-2 10 100 0.1 2 0 5 5 0.6 629.7 160.1 897.5 #10-06-3 10 100 0.1 3 0 5 5 0.6 905.8 179.5 1297.8 #13-00-1 13 100 0.1 1 0 0 5 0 224.7 153.1 395.7 #13-00-2 13 100 0.1 2 0 0 5 0 407.0 147.2 690.2 #13-00-3 13 100 0.1 3 0 0 5 0 594.7 161.3 1008.2 #13-02-1 13 100 0.1 1 0 0 5 0.2 227.4 127.2 384.6 #13-02-2 13 100 0.1 2 0 0 5 0.2 419.2 133.8 701.8 #13-02-3 13 100 0.1 3 0 0 5 0.2 610.0 147.0 1015.5 #13-04-1 13 100 0.1 1 0 0 5 0.4 290.9 149.4 454.9 #13-04-2 13 100 0.1 2 0 0 5 0.4 569.0 139.5 810.3 #13-04-3 13 100 0.1 3 0 0 5 0.4 818.5 140.4 1164.1 #13-06-1 13 100 0.1 1 0 0 5 0.6 308.3 127.0 447.8 #13-06-2 13 100 0.1 2 0 0 5 0.6 523.2 111.6 784.6 #13-06-3 13 100 0.1 3 0 0 5 0.6 825.9 112.1 1179.9 #15-00-1 15 100 0.1 1 0 0 0 0 246.9 145.8 394.6 #15-00-2 15 100 0.1 2 0 0 0 0 443.5 156.5 713.3 #15-00-3 15 100 0.1 3 0 0 0 0 663.8 166.4 1071.7 #15-02-1 15 100 0.1 1 0 0 0 0.2 294.6 167.1 451.6 #15-02-2 15 100 0.1 2 0 0 0 0.2 526.3 147.8 785.2 #15-02-3 15 100 0.1 3 0 0 0 0.2 749.3 185.3 1132.0 #15-04-1 15 100 0.1 1 0 0 0 0.4 371.4 182.1 529.4 #15-04-2 15 100 0.1 2 0 0 0 0.4 632.9 181.2 848.6 #15-04-3 15 100 0.1 3 0 0 0 0.4 887.3 202.5 1224.2 #15-06-1 15 100 0.1 1 0 0 0 0.6 360.3 154.6 490.5 #15-06-2 15 100 0.1 2 0 0 0 0.6 710.3 155.6 975.0 #15-06-3 15 100 0.1 3 0 0 0 0.6 1028.5 180.8 1408.7 #1-00-1 1 100 0.17 1 0 -5 0 0 337.1 270.5 608.6 #1-00-2 1 100 0.17 2 0 -5 0 0 718.9 328.3 1157.2 #1-00-3 1 100 0.17 3 0 -5 0 0 864.3 298.3 1315.3 #1-02-1 1 100 0.17 1 0 -5 0 0.2 352.4 257.0 623.8 #1-02-2 1 100 0.17 2 0 -5 0 0.2 777.4 299.2 1208.8 #1-02-3 1 100 0.17 3 0 -5 0 0.2 1166.6 351.1 1748.0 #1-04-1 1 100 0.17 1 0 -5 0 0.4 396.4 264.0 671.8 #1-04-2 1 100 0.17 2 0 -5 0 0.4 806.8 299.4 1256.7 #1-04-3 1 100 0.17 3 0 -5 0 0.4 1246.5 352.8 1868.5 #1-06-1 1 100 0.17 1 0 -5 0 0.6 399.8 263.7 677.7 #1-06-2 1 100 0.17 2 0 -5 0 0.6 818.5 298.5 1250.4
A10
#1-06-3 1 100 0.17 3 0 -5 0 0.6 1309.0 378.1 1914.2 #2-00-1 2 100 0.17 1 0 -5 5 0 272.5 233.2 567.5 #2-00-2 2 100 0.17 2 0 -5 5 0 568.1 260.4 1061.5 #2-00-3 2 100 0.17 3 0 -5 5 0 894.8 308.2 1579.4 #2-02-1 2 100 0.17 1 0 -5 5 0.2 315.9 237.8 615.7 #2-02-2 2 100 0.17 2 0 -5 5 0.2 626.1 250.8 1105.9 #2-02-3 2 100 0.17 3 0 -5 5 0.2 958.7 282.7 1633.1 #2-04-1 2 100 0.17 1 0 -5 5 0.4 315.6 230.7 590.7 #2-04-2 2 100 0.17 2 0 -5 5 0.4 643.5 250.6 1133.3 #2-04-3 2 100 0.17 3 0 -5 5 0.4 988.7 285.8 1674.3 #2-06-1 2 100 0.17 1 0 -5 5 0.6 390.3 242.0 650.8 #2-06-2 2 100 0.17 2 0 -5 5 0.6 991.0 308.3 1313.4 #2-06-3 2 100 0.17 3 0 -5 5 0.6 1363.2 335.9 1852.2 #3-00-1 3 100 0.17 1 0 -5 -5 0 410.8 289.6 683.0 #3-00-2 3 100 0.17 2 0 -5 -5 0 879.4 378.6 1268.7 #3-00-3 3 100 0.17 3 0 -5 -5 0 1299.9 411.0 1817.2 #3-02-1 3 100 0.17 1 0 -5 -5 0.2 413.9 275.6 668.2 #3-02-2 3 100 0.17 2 0 -5 -5 0.2 859.1 320.1 1254.6 #3-02-3 3 100 0.17 3 0 -5 -5 0.2 1304.6 388.8 1825.4 #3-04-1 3 100 0.17 1 0 -5 -5 0.4 471.7 301.6 723.0 #3-04-2 3 100 0.17 2 0 -5 -5 0.4 956.1 335.1 1336.2 #3-04-3 3 100 0.17 3 0 -5 -5 0.4 1473.9 394.6 1943.2 #3-06-1 3 100 0.17 1 0 -5 -5 0.6 511.6 332.0 742.1 #3-06-2 3 100 0.17 2 0 -5 -5 0.6 993.8 363.3 1351.8 #3-06-3 3 100 0.17 3 0 -5 -5 0.6 1526.2 414.5 2013.9 #4-00-1 4 100 0.17 1 0 5 -5 0 356.5 195.5 627.1 #4-00-2 4 100 0.17 2 0 5 -5 0 819.3 212.4 1226.7 #4-00-3 4 100 0.17 3 0 5 -5 0 1273.2 210.0 1807.4 #4-02-1 4 100 0.17 1 0 5 -5 0.2 394.4 224.9 663.1 #4-02-2 4 100 0.17 2 0 5 -5 0.2 841.5 234.3 1231.2 #4-02-3 4 100 0.17 3 0 5 -5 0.2 0.0 0.0 0.0 #4-04-1 4 100 0.17 1 0 5 -5 0.4 420.4 223.5 684.7 #4-04-2 4 100 0.17 2 0 5 -5 0.4 897.6 234.0 1274.6 #4-04-3 4 100 0.17 3 0 5 -5 0.4 1366.4 237.7 1850.1 #4-06-1 4 100 0.17 1 0 5 -5 0.6 423.0 222.3 690.3 #4-06-2 4 100 0.17 2 0 5 -5 0.6 920.0 245.6 1288.2 #4-06-3 4 100 0.17 3 0 5 -5 0.6 1337.4 229.4 1861.0 #9-00-1 9 100 0.17 1 0 5 0 0 337.4 203.9 612.9 #9-00-2 9 100 0.17 2 0 5 0 0 807.5 220.7 1223.3 #9-00-3 9 100 0.17 3 0 5 0 0 1195.6 245.8 1748.5 #9-02-1 9 100 0.17 1 0 5 0 0.2 364.0 246.4 660.7 #9-02-2 9 100 0.17 2 0 5 0 0.2 806.4 254.9 1254.8 #9-02-3 9 100 0.17 3 0 5 0 0.2 1232.6 279.8 1803.9 #9-04-1 9 100 0.17 1 0 5 0 0.4 438.3 264.3 734.0 #9-04-2 9 100 0.17 2 0 5 0 0.4 908.7 278.9 1330.6 #9-04-3 9 100 0.17 3 0 5 0 0.4 1325.5 290.6 1911.3 #9-06-1 9 100 0.17 1 0 5 0 0.6 470.8 266.1 757.4 #9-06-2 9 100 0.17 2 0 5 0 0.6 962.5 270.6 1387.6 #9-06-3 9 100 0.17 3 0 5 0 0.6 1416.8 260.9 2036.0
#10-00-1 10 100 0.17 1 0 5 5 0 275.9 196.7 567.1 #10-00-2 10 100 0.17 2 0 5 5 0 612.8 186.6 1066.4 #10-00-3 10 100 0.17 3 0 5 5 0 945.1 191.1 1601.1 #10-02-1 10 100 0.17 1 0 5 5 0.2 340.7 225.0 644.7 #10-02-2 10 100 0.17 2 0 5 5 0.2 706.4 221.4 1183.5 #10-02-3 10 100 0.17 3 0 5 5 0.2 1070.5 232.2 1716.3 #10-04-1 10 100 0.17 1 0 5 5 0.4 335.4 188.6 618.9
A11
#10-04-2 10 100 0.17 2 0 5 5 0.4 694.2 181.9 1161.2 #10-04-3 10 100 0.17 3 0 5 5 0.4 1035.6 154.8 1682.9 #10-06-1 10 100 0.17 1 0 5 5 0.6 425.2 257.1 723.2 #10-06-2 10 100 0.17 2 0 5 5 0.6 908.0 272.1 1370.1 #10-06-3 10 100 0.17 3 0 5 5 0.6 1295.1 279.8 1997.1 #13-00-1 13 100 0.17 1 0 0 5 0 292.1 227.8 582.1 #13-00-2 13 100 0.17 2 0 0 5 0 633.4 259.8 1109.0 #13-00-3 13 100 0.17 3 0 0 5 0 943.2 291.1 1621.9 #13-02-1 13 100 0.17 1 0 0 5 0.2 294.1 212.2 584.0 #13-02-2 13 100 0.17 2 0 0 5 0.2 605.8 222.6 1077.3 #13-02-3 13 100 0.17 3 0 0 5 0.2 992.4 268.5 1636.6 #13-04-1 13 100 0.17 1 0 0 5 0.4 358.1 222.9 639.8 #13-04-2 13 100 0.17 2 0 0 5 0.4 765.6 226.9 1201.4 #13-04-3 13 100 0.17 3 0 0 5 0.4 1121.3 242.2 1736.6 #13-06-1 13 100 0.17 1 0 0 5 0.6 361.7 203.8 657.7 #13-06-2 13 100 0.17 2 0 0 5 0.6 739.1 214.8 1201.1 #13-06-3 13 100 0.17 3 0 0 5 0.6 1156.4 224.1 1799.9 #15-00-1 15 100 0.17 1 0 0 0 0 323.8 247.1 602.4 #15-00-2 15 100 0.17 2 0 0 0 0 717.6 272.6 1157.2 #15-00-3 15 100 0.17 3 0 0 0 0 1069.9 298.7 1680.3 #15-02-1 15 100 0.17 1 0 0 0 0.2 372.5 242.9 673.7 #15-02-2 15 100 0.17 2 0 0 0 0.2 787.4 288.8 1253.6 #15-02-3 15 100 0.17 3 0 0 0 0.2 1151.2 318.1 1753.2 #15-04-1 15 100 0.17 1 0 0 0 0.4 461.7 264.3 693.2 #15-04-2 15 100 0.17 2 0 0 0 0.4 917.8 296.9 1322.0 #15-04-3 15 100 0.17 3 0 0 0 0.4 1342.1 357.7 1906.7 #15-06-1 15 100 0.17 1 0 0 0 0.6 432.6 227.5 699.5 #15-06-2 15 100 0.17 2 0 0 0 0.6 929.9 268.6 1380.1 #15-06-3 15 100 0.17 3 0 0 0 0.6 1398.7 301.5 2025.0 #1-00-1 1 100 0.24 1 0 -5 0 0 401.0 378.7 811.5 #1-00-2 1 100 0.24 2 0 -5 0 0 896.8 450.0 1550.1 #1-00-3 1 100 0.24 3 0 -5 0 0 1387.9 544.0 2263.6 #1-02-1 1 100 0.24 1 0 -5 0 0.2 408.6 362.9 805.6 #1-02-2 1 100 0.24 2 0 -5 0 0.2 995.8 438.6 1660.9 #1-02-3 1 100 0.24 3 0 -5 0 0.2 1479.2 499.8 2339.3 #1-04-1 1 100 0.24 1 0 -5 0 0.4 449.8 369.6 872.4 #1-04-2 1 100 0.24 2 0 -5 0 0.4 1030.4 450.2 1700.7 #1-04-3 1 100 0.24 3 0 -5 0 0.4 1523.9 499.0 2431.6 #1-06-1 1 100 0.24 1 0 -5 0 0.6 448.5 366.7 858.4 #1-06-2 1 100 0.24 2 0 -5 0 0.6 1036.1 448.4 1717.6 #1-06-3 1 100 0.24 3 0 -5 0 0.6 1469.8 469.5 2384.0 #2-00-1 2 100 0.24 1 0 -5 5 0 333.3 335.0 760.0 #2-00-2 2 100 0.24 2 0 -5 5 0 814.4 403.1 1492.6 #2-00-3 2 100 0.24 3 0 -5 5 0 1140.5 431.1 2102.5 #2-02-1 2 100 0.24 1 0 -5 5 0.2 375.0 339.9 801.9 #2-02-2 2 100 0.24 2 0 -5 5 0.2 842.0 395.7 1558.6 #2-02-3 2 100 0.24 3 0 -5 5 0.2 1265.7 424.6 2222.3 #2-04-1 2 100 0.24 1 0 -5 5 0.4 363.4 322.0 766.7 #2-04-2 2 100 0.24 2 0 -5 5 0.4 819.2 359.5 1517.5 #2-04-3 2 100 0.24 3 0 -5 5 0.4 1311.1 424.1 2265.3 #2-06-1 2 100 0.24 1 0 -5 5 0.6 447.9 341.5 841.4 #2-06-2 2 100 0.24 2 0 -5 5 0.6 1119.4 419.8 1644.9 #2-06-3 2 100 0.24 3 0 -5 5 0.6 1560.6 443.7 2388.5 #3-00-1 3 100 0.24 1 0 -5 -5 0 510.9 418.5 912.0 #3-00-2 3 100 0.24 2 0 -5 -5 0 1113.0 535.1 1685.3 #3-00-3 3 100 0.24 3 0 -5 -5 0 1597.4 566.8 2416.1
A12
#3-02-1 3 100 0.24 1 0 -5 -5 0.2 501.9 382.1 897.5 #3-02-2 3 100 0.24 2 0 -5 -5 0.2 1131.3 483.5 1714.8 #3-02-3 3 100 0.24 3 0 -5 -5 0.2 1657.3 553.1 2405.2 #3-04-1 3 100 0.24 1 0 -5 -5 0.4 559.7 414.1 959.2 #3-04-2 3 100 0.24 2 0 -5 -5 0.4 1228.3 507.4 1826.8 #3-04-3 3 100 0.24 3 0 -5 -5 0.4 1753.8 544.8 2548.9 #3-06-1 3 100 0.24 1 0 -5 -5 0.6 597.2 438.6 969.7 #3-06-2 3 100 0.24 2 0 -5 -5 0.6 1285.0 523.6 1844.0 #3-06-3 3 100 0.24 3 0 -5 -5 0.6 1785.8 565.4 2577.2 #4-00-1 4 100 0.24 1 0 5 -5 0 429.7 276.0 827.1 #4-00-2 4 100 0.24 2 0 5 -5 0 1088.6 305.7 1664.6 #4-00-3 4 100 0.24 3 0 5 -5 0 1597.6 333.0 2427.2 #4-02-1 4 100 0.24 1 0 5 -5 0.2 466.3 293.7 873.1 #4-02-2 4 100 0.24 2 0 5 -5 0.2 1040.5 328.5 1639.1 #4-02-3 4 100 0.24 3 0 5 -5 0.2 0.0 0.0 0.0 #4-04-1 4 100 0.24 1 0 5 -5 0.4 510.9 307.2 914.8 #4-04-2 4 100 0.24 2 0 5 -5 0.4 1170.2 327.8 1724.0 #4-04-3 4 100 0.24 3 0 5 -5 0.4 1724.2 311.4 2486.7 #4-06-1 4 100 0.24 1 0 5 -5 0.6 482.0 293.3 895.2 #4-06-2 4 100 0.24 2 0 5 -5 0.6 1158.6 339.1 1743.7 #4-06-3 4 100 0.24 3 0 5 -5 0.6 1804.7 369.0 2547.4 #9-00-1 9 100 0.24 1 0 5 0 0 395.6 284.5 796.6 #9-00-2 9 100 0.24 2 0 5 0 0 1035.9 340.7 1675.8 #9-00-3 9 100 0.24 3 0 5 0 0 1483.1 317.8 2301.5 #9-02-1 9 100 0.24 1 0 5 0 0.2 410.0 324.3 827.2 #9-02-2 9 100 0.24 2 0 5 0 0.2 1031.4 362.5 1701.6 #9-02-3 9 100 0.24 3 0 5 0 0.2 1818.6 358.1 2586.6 #9-04-1 9 100 0.24 1 0 5 0 0.4 505.9 359.9 957.5 #9-04-2 9 100 0.24 2 0 5 0 0.4 1088.8 366.7 1742.3 #9-04-3 9 100 0.24 3 0 5 0 0.4 1611.1 374.0 2496.1 #9-06-1 9 100 0.24 1 0 5 0 0.6 521.4 354.7 951.0 #9-06-2 9 100 0.24 2 0 5 0 0.6 1149.5 359.6 1851.4 #9-06-3 9 100 0.24 3 0 5 0 0.6 1748.6 376.5 2651.6
#10-00-1 10 100 0.24 1 0 5 5 0 319.6 267.1 731.9 #10-00-2 10 100 0.24 2 0 5 5 0 866.1 279.1 1498.9 #10-00-3 10 100 0.24 3 0 5 5 0 1228.8 282.7 2158.7 #10-02-1 10 100 0.24 1 0 5 5 0.2 388.0 302.5 809.0 #10-02-2 10 100 0.24 2 0 5 5 0.2 900.9 319.8 1606.4 #10-02-3 10 100 0.24 3 0 5 5 0.2 1349.5 317.0 2304.7 #10-04-1 10 100 0.24 1 0 5 5 0.4 374.6 280.6 798.4 #10-04-2 10 100 0.24 2 0 5 5 0.4 927.9 281.3 1599.6 #10-04-3 10 100 0.24 3 0 5 5 0.4 1405.7 259.3 2386.1 #10-06-1 10 100 0.24 1 0 5 5 0.6 469.1 339.0 906.7 #10-06-2 10 100 0.24 2 0 5 5 0.6 1101.0 359.4 1834.6 #10-06-3 10 100 0.24 3 0 5 5 0.6 1583.8 367.1 2586.6 #13-00-1 13 100 0.24 1 0 0 5 0 361.7 321.2 771.8 #13-00-2 13 100 0.24 2 0 0 5 0 819.9 367.3 1504.6 #13-00-3 13 100 0.24 3 0 0 5 0 1203.0 381.7 2153.3 #13-02-1 13 100 0.24 1 0 0 5 0.2 354.4 291.9 770.9 #13-02-2 13 100 0.24 2 0 0 5 0.2 789.6 327.1 1464.8 #13-02-3 13 100 0.24 3 0 0 5 0.2 1205.9 364.8 2153.5 #13-04-1 13 100 0.24 1 0 0 5 0.4 403.1 302.4 814.1 #13-04-2 13 100 0.24 2 0 0 5 0.4 951.6 343.2 1598.5 #13-04-3 13 100 0.24 3 0 0 5 0.4 1404.5 337.9 2288.2 #13-06-1 13 100 0.24 1 0 0 5 0.6 392.2 301.8 812.3 #13-06-2 13 100 0.24 2 0 0 5 0.6 911.7 303.8 1578.8
A13
#13-06-3 13 100 0.24 3 0 0 5 0.6 1338.6 307.2 2293.9 #15-00-1 15 100 0.24 1 0 0 0 0 399.7 341.8 793.9 #15-00-2 15 100 0.24 2 0 0 0 0 922.4 395.3 1608.0 #15-00-3 15 100 0.24 3 0 0 0 0 1348.0 440.7 2244.2 #15-02-1 15 100 0.24 1 0 0 0 0.2 434.8 347.8 844.8 #15-02-2 15 100 0.24 2 0 0 0 0.2 1019.5 409.5 1696.5 #15-02-3 15 100 0.24 3 0 0 0 0.2 1452.5 435.0 2324.5 #15-04-1 15 100 0.24 1 0 0 0 0.4 515.7 373.0 904.1 #15-04-2 15 100 0.24 2 0 0 0 0.4 1136.6 413.6 1767.7 #15-04-3 15 100 0.24 3 0 0 0 0.4 1567.2 457.1 2429.6 #15-06-1 15 100 0.24 1 0 0 0 0.6 491.6 323.3 914.8 #15-06-2 15 100 0.24 2 0 0 0 0.6 1114.9 364.7 1814.0 #15-06-3 15 100 0.24 3 0 0 0 0.6 1548.2 379.8 2515.1 #1-00-1 1 200 0.1 1 0 -5 0 0 243.9 166.4 394.6 #1-00-2 1 200 0.1 2 0 -5 0 0 507.5 218.1 735.3 #1-00-3 1 200 0.1 3 0 -5 0 0 746.6 241.6 1065.7 #1-02-1 1 200 0.1 1 0 -5 0 0.2 256.0 156.1 403.1 #1-02-2 1 200 0.1 2 0 -5 0 0.2 529.6 183.8 775.9 #1-02-3 1 200 0.1 3 0 -5 0 0.2 781.2 217.1 1150.0 #1-04-1 1 200 0.1 1 0 -5 0 0.4 267.7 164.6 424.6 #1-04-2 1 200 0.1 2 0 -5 0 0.4 525.8 192.6 788.2 #1-04-3 1 200 0.1 3 0 -5 0 0.4 791.5 251.3 1226.2 #1-06-1 1 200 0.1 1 0 -5 0 0.6 270.6 160.2 421.8 #1-06-2 1 200 0.1 2 0 -5 0 0.6 539.0 191.8 795.8 #1-06-3 1 200 0.1 3 0 -5 0 0.6 849.7 227.9 1214.6 #2-00-1 2 200 0.1 1 0 -5 5 0 202.7 153.8 364.2 #2-00-2 2 200 0.1 2 0 -5 5 0 424.9 183.4 689.4 #2-00-3 2 200 0.1 3 0 -5 5 0 621.7 200.0 1001.0 #2-02-1 2 200 0.1 1 0 -5 5 0.2 230.0 149.0 397.5 #2-02-2 2 200 0.1 2 0 -5 5 0.2 465.7 164.1 744.1 #2-02-3 2 200 0.1 3 0 -5 5 0.2 685.6 180.8 1087.4 #2-04-1 2 200 0.1 1 0 -5 5 0.4 231.7 152.0 384.6 #2-04-2 2 200 0.1 2 0 -5 5 0.4 467.7 167.6 741.1 #2-04-3 2 200 0.1 3 0 -5 5 0.4 685.7 186.7 1093.4 #2-06-1 2 200 0.1 1 0 -5 5 0.6 669.3 207.8 554.8 #2-06-2 2 200 0.1 2 0 -5 5 0.6 1063.8 269.2 931.5 #2-06-3 2 200 0.1 3 0 -5 5 0.6 1315.6 308.5 1305.8 #3-00-1 3 200 0.1 1 0 -5 -5 0 286.1 181.5 420.2 #3-00-2 3 200 0.1 2 0 -5 -5 0 556.0 225.8 763.3 #3-00-3 3 200 0.1 3 0 -5 -5 0 823.5 254.0 1109.6 #3-02-1 3 200 0.1 1 0 -5 -5 0.2 291.0 164.7 414.7 #3-02-2 3 200 0.1 2 0 -5 -5 0.2 561.7 197.1 771.1 #3-02-3 3 200 0.1 3 0 -5 -5 0.2 853.3 236.2 1161.2 #3-04-1 3 200 0.1 1 0 -5 -5 0.4 321.9 174.6 450.5 #3-04-2 3 200 0.1 2 0 -5 -5 0.4 611.0 205.6 837.0 #3-04-3 3 200 0.1 3 0 -5 -5 0.4 939.5 229.8 1269.7 #3-06-1 3 200 0.1 1 0 -5 -5 0.6 338.0 193.1 471.2 #3-06-2 3 200 0.1 2 0 -5 -5 0.6 613.9 212.0 842.6 #3-06-3 3 200 0.1 3 0 -5 -5 0.6 930.0 242.2 1262.2 #4-00-1 4 200 0.1 1 0 5 -5 0 268.6 133.5 403.9 #4-00-2 4 200 0.1 2 0 5 -5 0 549.3 138.9 757.0 #4-00-3 4 200 0.1 3 0 5 -5 0 827.7 136.8 1122.4 #4-02-1 4 200 0.1 1 0 5 -5 0.2 282.0 155.3 424.5 #4-02-2 4 200 0.1 2 0 5 -5 0.2 538.4 139.7 766.5 #4-02-3 4 200 0.1 3 0 5 -5 0.2 0.0 0.0 0.0 #4-04-1 4 200 0.1 1 0 5 -5 0.4 299.2 144.0 435.3
A14
#4-04-2 4 200 0.1 2 0 5 -5 0.4 567.4 136.2 785.6 #4-04-3 4 200 0.1 3 0 5 -5 0.4 841.0 135.8 1146.4 #4-06-1 4 200 0.1 1 0 5 -5 0.6 302.3 151.4 437.3 #4-06-2 4 200 0.1 2 0 5 -5 0.6 565.8 147.9 785.4 #4-06-3 4 200 0.1 3 0 5 -5 0.6 867.2 135.6 1198.7 #9-00-1 9 200 0.1 1 0 5 0 0 252.5 134.0 392.4 #9-00-2 9 200 0.1 2 0 5 0 0 513.1 132.2 735.8 #9-00-3 9 200 0.1 3 0 5 0 0 735.7 128.7 1052.6 #9-02-1 9 200 0.1 1 0 5 0 0.2 263.5 167.6 430.2 #9-02-2 9 200 0.1 2 0 5 0 0.2 538.5 160.6 805.3 #9-02-3 9 200 0.1 3 0 5 0 0.2 757.9 176.1 1135.1 #9-04-1 9 200 0.1 1 0 5 0 0.4 313.1 166.4 491.8 #9-04-2 9 200 0.1 2 0 5 0 0.4 582.2 170.0 859.1 #9-04-3 9 200 0.1 3 0 5 0 0.4 821.7 175.5 1192.3 #9-06-1 9 200 0.1 1 0 5 0 0.6 314.8 155.8 483.3 #9-06-2 9 200 0.1 2 0 5 0 0.6 577.3 155.0 874.3 #9-06-3 9 200 0.1 3 0 5 0 0.6 810.7 151.1 1218.3
#10-00-1 10 200 0.1 1 0 5 5 0 219.0 132.3 376.1 #10-00-2 10 200 0.1 2 0 5 5 0 449.6 135.1 701.5 #10-00-3 10 200 0.1 3 0 5 5 0 658.7 132.0 1025.0 #10-02-1 10 200 0.1 1 0 5 5 0.2 264.9 155.0 440.9 #10-02-2 10 200 0.1 2 0 5 5 0.2 503.3 150.2 787.2 #10-02-3 10 200 0.1 3 0 5 5 0.2 730.8 147.9 1134.3 #10-04-1 10 200 0.1 1 0 5 5 0.4 247.8 126.3 405.7 #10-04-2 10 200 0.1 2 0 5 5 0.4 478.9 113.8 748.2 #10-04-3 10 200 0.1 3 0 5 5 0.4 681.5 105.7 1075.9 #10-06-1 10 200 0.1 1 0 5 5 0.6 293.8 163.4 478.0 #10-06-2 10 200 0.1 2 0 5 5 0.6 543.5 151.9 856.0 #10-06-3 10 200 0.1 3 0 5 5 0.6 737.6 169.8 1184.8 #13-00-1 13 200 0.1 1 0 0 5 0 232.7 161.5 388.1 #13-00-2 13 200 0.1 2 0 0 5 0 441.6 169.8 694.2 #13-00-3 13 200 0.1 3 0 0 5 0 654.6 181.7 1016.4 #13-02-1 13 200 0.1 1 0 0 5 0.2 226.8 142.7 391.7 #13-02-2 13 200 0.1 2 0 0 5 0.2 446.0 153.1 718.1 #13-02-3 13 200 0.1 3 0 0 5 0.2 657.4 161.5 1051.4 #13-04-1 13 200 0.1 1 0 0 5 0.4 261.0 139.6 414.1 #13-04-2 13 200 0.1 2 0 0 5 0.4 679.0 152.4 798.8 #13-04-3 13 200 0.1 3 0 0 5 0.4 899.8 164.0 1140.2 #13-06-1 13 200 0.1 1 0 0 5 0.6 244.1 130.7 405.7 #13-06-2 13 200 0.1 2 0 0 5 0.6 485.7 132.4 761.0 #13-06-3 13 200 0.1 3 0 0 5 0.6 698.9 149.8 1119.9 #15-00-1 15 200 0.1 1 0 0 0 0 244.7 159.4 385.8 #15-00-2 15 200 0.1 2 0 0 0 0 492.8 175.0 724.4 #15-00-3 15 200 0.1 3 0 0 0 0 718.6 199.1 1060.5 #15-02-1 15 200 0.1 1 0 0 0 0.2 267.4 165.3 430.2 #15-02-2 15 200 0.1 2 0 0 0 0.2 520.2 181.2 787.2 #15-02-3 15 200 0.1 3 0 0 0 0.2 760.9 201.7 1140.8 #15-04-1 15 200 0.1 1 0 0 0 0.4 343.5 183.4 464.5 #15-04-2 15 200 0.1 2 0 0 0 0.4 593.7 197.3 824.8 #15-04-3 15 200 0.1 3 0 0 0 0.4 834.1 217.1 1177.0 #15-06-1 15 200 0.1 1 0 0 0 0.6 291.6 150.8 460.2 #15-06-2 15 200 0.1 2 0 0 0 0.6 593.9 148.9 902.5 #15-06-3 15 200 0.1 3 0 0 0 0.6 911.3 160.9 1353.0 #1-00-1 1 200 0.17 1 0 -5 0 0 300.0 253.0 580.0 #1-00-2 1 200 0.17 2 0 -5 0 0 648.9 310.5 1088.5 #1-00-3 1 200 0.17 3 0 -5 0 0 999.5 378.1 1618.9
A15
#1-02-1 1 200 0.17 1 0 -5 0 0.2 301.9 243.5 581.3 #1-02-2 1 200 0.17 2 0 -5 0 0.2 670.1 289.8 1149.7 #1-02-3 1 200 0.17 3 0 -5 0 0.2 1025.4 334.2 1687.6 #1-04-1 1 200 0.17 1 0 -5 0 0.4 314.7 252.6 603.4 #1-04-2 1 200 0.17 2 0 -5 0 0.4 674.9 292.6 1164.0 #1-04-3 1 200 0.17 3 0 -5 0 0.4 1073.1 374.7 1776.9 #1-06-1 1 200 0.17 1 0 -5 0 0.6 314.7 248.1 603.1 #1-06-2 1 200 0.17 2 0 -5 0 0.6 662.9 274.6 1137.6 #1-06-3 1 200 0.17 3 0 -5 0 0.6 1023.9 345.0 1727.9 #2-00-1 2 200 0.17 1 0 -5 5 0 249.2 232.7 547.5 #2-00-2 2 200 0.17 2 0 -5 5 0 547.9 265.8 1038.2 #2-00-3 2 200 0.17 3 0 -5 5 0 812.2 303.5 1508.7 #2-02-1 2 200 0.17 1 0 -5 5 0.2 276.0 226.6 578.3 #2-02-2 2 200 0.17 2 0 -5 5 0.2 592.1 242.8 1093.4 #2-02-3 2 200 0.17 3 0 -5 5 0.2 864.7 285.3 1623.3 #2-04-1 2 200 0.17 1 0 -5 5 0.4 270.8 226.2 559.7 #2-04-2 2 200 0.17 2 0 -5 5 0.4 583.3 250.0 1086.5 #2-04-3 2 200 0.17 3 0 -5 5 0.4 944.6 281.6 1833.4 #2-06-1 2 200 0.17 1 0 -5 5 0.6 761.0 296.5 791.6 #2-06-2 2 200 0.17 2 0 -5 5 0.6 1162.0 359.3 1291.7 #2-06-3 2 200 0.17 3 0 -5 5 0.6 1485.7 410.3 1807.5 #3-00-1 3 200 0.17 1 0 -5 -5 0 365.8 270.0 629.7 #3-00-2 3 200 0.17 2 0 -5 -5 0 746.8 333.5 1157.1 #3-00-3 3 200 0.17 3 0 -5 -5 0 1115.2 386.3 1687.6 #3-02-1 3 200 0.17 1 0 -5 -5 0.2 373.0 255.3 637.8 #3-02-2 3 200 0.17 2 0 -5 -5 0.2 743.7 307.3 1157.0 #3-02-3 3 200 0.17 3 0 -5 -5 0.2 1143.3 393.4 1759.5 #3-04-1 3 200 0.17 1 0 -5 -5 0.4 393.8 254.8 663.7 #3-04-2 3 200 0.17 2 0 -5 -5 0.4 813.5 316.3 1257.7 #3-04-3 3 200 0.17 3 0 -5 -5 0.4 1221.1 366.9 1859.6 #3-06-1 3 200 0.17 1 0 -5 -5 0.6 398.2 266.2 675.8 #3-06-2 3 200 0.17 2 0 -5 -5 0.6 821.2 316.2 1276.9 #3-06-3 3 200 0.17 3 0 -5 -5 0.6 1179.3 375.4 1834.5 #4-00-1 4 200 0.17 1 0 5 -5 0 339.4 199.2 599.4 #4-00-2 4 200 0.17 2 0 5 -5 0 721.5 202.3 1134.8 #4-00-3 4 200 0.17 3 0 5 -5 0 1114.6 209.9 1692.0 #4-02-1 4 200 0.17 1 0 5 -5 0.2 337.2 211.7 619.7 #4-02-2 4 200 0.17 2 0 5 -5 0.2 706.0 201.8 1149.1 #4-02-3 4 200 0.17 3 0 5 -5 0.2 0.0 0.0 0.0 #4-04-1 4 200 0.17 1 0 5 -5 0.4 364.3 187.1 637.2 #4-04-2 4 200 0.17 2 0 5 -5 0.4 754.2 212.5 1174.0 #4-04-3 4 200 0.17 3 0 5 -5 0.4 1113.7 189.1 1732.6 #4-06-1 4 200 0.17 1 0 5 -5 0.6 353.8 198.6 640.7 #4-06-2 4 200 0.17 2 0 5 -5 0.6 719.7 186.0 1225.8 #4-06-3 4 200 0.17 3 0 5 -5 0.6 1116.8 197.6 1764.8 #9-00-1 9 200 0.17 1 0 5 0 0 304.1 203.5 567.0 #9-00-2 9 200 0.17 2 0 5 0 0 649.5 183.9 1073.8 #9-00-3 9 200 0.17 3 0 5 0 0 953.2 180.9 1568.8 #9-02-1 9 200 0.17 1 0 5 0 0.2 309.2 222.1 620.2 #9-02-2 9 200 0.17 2 0 5 0 0.2 689.6 218.7 1170.1 #9-02-3 9 200 0.17 3 0 5 0 0.2 1294.1 263.6 1878.3 #9-04-1 9 200 0.17 1 0 5 0 0.4 367.6 238.1 678.3 #9-04-2 9 200 0.17 2 0 5 0 0.4 722.0 230.1 1209.4 #9-04-3 9 200 0.17 3 0 5 0 0.4 1044.2 255.4 1752.5 #9-06-1 9 200 0.17 1 0 5 0 0.6 352.1 223.7 654.4 #9-06-2 9 200 0.17 2 0 5 0 0.6 692.2 221.1 1209.8
A16
#9-06-3 9 200 0.17 3 0 5 0 0.6 1005.5 201.4 1717.5 #10-00-1 10 200 0.17 1 0 5 5 0 264.0 190.5 540.6 #10-00-2 10 200 0.17 2 0 5 5 0 567.9 187.5 1042.1 #10-00-3 10 200 0.17 3 0 5 5 0 830.0 173.8 1503.5 #10-02-1 10 200 0.17 1 0 5 5 0.2 286.3 203.6 605.1 #10-02-2 10 200 0.17 2 0 5 5 0.2 610.7 202.4 1129.0 #10-02-3 10 200 0.17 3 0 5 5 0.2 901.9 198.6 1628.9 #10-04-1 10 200 0.17 1 0 5 5 0.4 285.4 191.0 575.2 #10-04-2 10 200 0.17 2 0 5 5 0.4 565.0 154.1 1055.1 #10-04-3 10 200 0.17 3 0 5 5 0.4 865.8 146.7 1573.4 #10-06-1 10 200 0.17 1 0 5 5 0.6 326.2 214.6 640.6 #10-06-2 10 200 0.17 2 0 5 5 0.6 638.0 216.5 1196.2 #10-06-3 10 200 0.17 3 0 5 5 0.6 899.8 192.1 1685.8 #13-00-1 13 200 0.17 1 0 0 5 0 286.0 230.4 566.4 #13-00-2 13 200 0.17 2 0 0 5 0 583.8 250.2 1080.9 #13-00-3 13 200 0.17 3 0 0 5 0 829.8 259.9 1526.1 #13-02-1 13 200 0.17 1 0 0 5 0.2 269.5 204.6 556.7 #13-02-2 13 200 0.17 2 0 0 5 0.2 536.9 211.3 1041.6 #13-02-3 13 200 0.17 3 0 0 5 0.2 798.7 237.2 1527.5 #13-04-1 13 200 0.17 1 0 0 5 0.4 301.1 200.9 584.9 #13-04-2 13 200 0.17 2 0 0 5 0.4 794.5 229.5 1152.1 #13-04-3 13 200 0.17 3 0 0 5 0.4 1030.8 224.6 1587.6 #13-06-1 13 200 0.17 1 0 0 5 0.6 273.5 187.2 556.0 #13-06-2 13 200 0.17 2 0 0 5 0.6 559.6 193.4 1053.1 #13-06-3 13 200 0.17 3 0 0 5 0.6 851.7 217.8 1615.5 #15-00-1 15 200 0.17 1 0 0 0 0 311.8 241.4 579.5 #15-00-2 15 200 0.17 2 0 0 0 0 630.9 253.4 1098.4 #15-00-3 15 200 0.17 3 0 0 0 0 991.7 300.4 1667.0 #15-02-1 15 200 0.17 1 0 0 0 0.2 308.5 232.8 591.7 #15-02-2 15 200 0.17 2 0 0 0 0.2 678.4 270.0 1176.0 #15-02-3 15 200 0.17 3 0 0 0 0.2 956.6 275.5 1659.2 #15-04-1 15 200 0.17 1 0 0 0 0.4 386.5 252.4 641.6 #15-04-2 15 200 0.17 2 0 0 0 0.4 723.4 278.5 1135.5 #15-04-3 15 200 0.17 3 0 0 0 0.4 1065.1 314.1 1706.5 #15-06-1 15 200 0.17 1 0 0 0 0.6 341.7 212.4 646.7 #15-06-2 15 200 0.17 2 0 0 0 0.6 739.6 220.6 1275.5 #15-06-3 15 200 0.17 3 0 0 0 0.6 1092.2 246.1 1863.0 #1-00-1 1 200 0.24 1 0 -5 0 0 337.1 323.5 733.2 #1-00-2 1 200 0.24 2 0 -5 0 0 765.5 409.4 1434.3 #1-00-3 1 200 0.24 3 0 -5 0 0 1201.9 492.4 2145.5 #1-02-1 1 200 0.24 1 0 -5 0 0.2 342.7 306.9 742.5 #1-02-2 1 200 0.24 2 0 -5 0 0.2 734.0 353.2 1449.4 #1-02-3 1 200 0.24 3 0 -5 0 0.2 1209.7 448.7 2201.9 #1-04-1 1 200 0.24 1 0 -5 0 0.4 349.9 306.8 767.4 #1-04-2 1 200 0.24 2 0 -5 0 0.4 812.3 409.8 1520.4 #1-04-3 1 200 0.24 3 0 -5 0 0.4 1279.3 453.0 2325.7 #1-06-1 1 200 0.24 1 0 -5 0 0.6 355.0 301.1 765.3 #1-06-2 1 200 0.24 2 0 -5 0 0.6 820.0 375.6 1498.5 #1-06-3 1 200 0.24 3 0 -5 0 0.6 1167.0 457.6 2233.1 #2-00-1 2 200 0.24 1 0 -5 5 0 293.7 300.9 712.8 #2-00-2 2 200 0.24 2 0 -5 5 0 625.4 348.9 1352.1 #2-00-3 2 200 0.24 3 0 -5 5 0 939.5 397.8 1977.2 #2-02-1 2 200 0.24 1 0 -5 5 0.2 309.7 292.9 741.9 #2-02-2 2 200 0.24 2 0 -5 5 0.2 688.1 331.0 1425.1 #2-02-3 2 200 0.24 3 0 -5 5 0.2 1079.8 369.9 2210.9 #2-04-1 2 200 0.24 1 0 -5 5 0.4 304.8 280.4 717.4
A17
#2-04-2 2 200 0.24 2 0 -5 5 0.4 648.4 332.9 1413.9 #2-04-3 2 200 0.24 3 0 -5 5 0.4 983.7 349.2 2059.0 #2-06-1 2 200 0.24 1 0 -5 5 0.6 814.3 375.0 963.3 #2-06-2 2 200 0.24 2 0 -5 5 0.6 1234.6 451.9 1582.2 #2-06-3 2 200 0.24 3 0 -5 5 0.6 1637.6 493.2 2204.6 #3-00-1 3 200 0.24 1 0 -5 -5 0 416.4 362.4 800.1 #3-00-2 3 200 0.24 2 0 -5 -5 0 892.9 437.1 1521.0 #3-00-3 3 200 0.24 3 0 -5 -5 0 1323.6 513.8 2206.2 #3-02-1 3 200 0.24 1 0 -5 -5 0.2 416.3 361.9 811.8 #3-02-2 3 200 0.24 2 0 -5 -5 0.2 850.4 400.3 1526.2 #3-02-3 3 200 0.24 3 0 -5 -5 0.2 1375.2 522.1 2297.0 #3-04-1 3 200 0.24 1 0 -5 -5 0.4 438.5 337.1 834.2 #3-04-2 3 200 0.24 2 0 -5 -5 0.4 954.9 402.8 1649.1 #3-04-3 3 200 0.24 3 0 -5 -5 0.4 1441.8 504.7 2420.1 #3-06-1 3 200 0.24 1 0 -5 -5 0.6 432.2 378.3 843.8 #3-06-2 3 200 0.24 2 0 -5 -5 0.6 965.9 432.5 1654.2 #3-06-3 3 200 0.24 3 0 -5 -5 0.6 1448.4 530.5 2415.0 #4-00-1 4 200 0.24 1 0 5 -5 0 369.4 252.3 755.5 #4-00-2 4 200 0.24 2 0 5 -5 0 843.5 262.7 1483.1 #4-00-3 4 200 0.24 3 0 5 -5 0 1341.9 265.4 2225.6 #4-02-1 4 200 0.24 1 0 5 -5 0.2 382.9 289.1 809.6 #4-02-2 4 200 0.24 2 0 5 -5 0.2 878.3 260.0 1546.1 #4-02-3 4 200 0.24 3 0 5 -5 0.2 0.0 0.0 0.0 #4-04-1 4 200 0.24 1 0 5 -5 0.4 359.5 260.9 790.3 #4-04-2 4 200 0.24 2 0 5 -5 0.4 875.2 252.3 1527.9 #4-04-3 4 200 0.24 3 0 5 -5 0.4 1374.0 254.6 2311.8 #4-06-1 4 200 0.24 1 0 5 -5 0.6 380.4 239.0 814.3 #4-06-2 4 200 0.24 2 0 5 -5 0.6 877.2 241.7 1604.4 #4-06-3 4 200 0.24 3 0 5 -5 0.6 1387.4 286.8 2309.6 #9-00-1 9 200 0.24 1 0 5 0 0 353.1 242.7 742.6 #9-00-2 9 200 0.24 2 0 5 0 0 895.0 250.2 1516.2 #9-00-3 9 200 0.24 3 0 5 0 0 1465.1 315.6 2323.4 #9-02-1 9 200 0.24 1 0 5 0 0.2 327.9 269.6 756.4 #9-02-2 9 200 0.24 2 0 5 0 0.2 842.0 279.1 1557.7 #9-02-3 9 200 0.24 3 0 5 0 0.2 1701.5 409.0 2619.2 #9-04-1 9 200 0.24 1 0 5 0 0.4 395.9 290.6 842.6 #9-04-2 9 200 0.24 2 0 5 0 0.4 926.1 303.1 1605.2 #9-04-3 9 200 0.24 3 0 5 0 0.4 1245.8 299.3 2286.3 #9-06-1 9 200 0.24 1 0 5 0 0.6 373.2 279.7 830.5 #9-06-2 9 200 0.24 2 0 5 0 0.6 771.0 260.8 1521.5 #9-06-3 9 200 0.24 3 0 5 0 0.6 1186.6 245.5 2217.3
#10-00-1 10 200 0.24 1 0 5 5 0 287.8 237.7 688.7 #10-00-2 10 200 0.24 2 0 5 5 0 670.3 252.4 1386.7 #10-00-3 10 200 0.24 3 0 5 5 0 1007.1 237.8 2015.9 #10-02-1 10 200 0.24 1 0 5 5 0.2 319.0 251.0 756.9 #10-02-2 10 200 0.24 2 0 5 5 0.2 705.6 261.3 1454.8 #10-02-3 10 200 0.24 3 0 5 5 0.2 1090.0 226.5 2127.1 #10-04-1 10 200 0.24 1 0 5 5 0.4 311.2 238.9 753.3 #10-04-2 10 200 0.24 2 0 5 5 0.4 688.5 199.0 1411.6 #10-04-3 10 200 0.24 3 0 5 5 0.4 999.9 202.6 2050.7 #10-06-1 10 200 0.24 1 0 5 5 0.6 342.7 254.0 800.0 #10-06-2 10 200 0.24 2 0 5 5 0.6 741.6 250.4 1529.4 #10-06-3 10 200 0.24 3 0 5 5 0.6 1058.6 261.8 2194.4 #13-00-1 13 200 0.24 1 0 0 5 0 316.3 294.0 724.3 #13-00-2 13 200 0.24 2 0 0 5 0 649.9 316.4 1369.2 #13-00-3 13 200 0.24 3 0 0 5 0 964.3 333.4 1997.0
A18
#13-02-1 13 200 0.24 1 0 0 5 0.2 287.9 250.1 721.8 #13-02-2 13 200 0.24 2 0 0 5 0.2 595.7 255.5 1322.3 #13-02-3 13 200 0.24 3 0 0 5 0.2 924.9 297.7 1991.1 #13-04-1 13 200 0.24 1 0 0 5 0.4 318.9 243.8 740.1 #13-04-2 13 200 0.24 2 0 0 5 0.4 857.8 287.2 1455.9 #13-04-3 13 200 0.24 3 0 0 5 0.4 1121.3 286.7 2041.9 #13-06-1 13 200 0.24 1 0 0 5 0.6 304.9 223.8 727.0 #13-06-2 13 200 0.24 2 0 0 5 0.6 610.8 247.7 1359.9 #13-06-3 13 200 0.24 3 0 0 5 0.6 1011.3 262.9 2101.0 #15-00-1 15 200 0.24 1 0 0 0 0 340.8 308.4 747.6 #15-00-2 15 200 0.24 2 0 0 0 0 747.3 322.9 1458.3 #15-00-3 15 200 0.24 3 0 0 0 0 1070.5 377.2 2082.8 #15-02-1 15 200 0.24 1 0 0 0 0.2 331.2 277.6 738.8 #15-02-2 15 200 0.24 2 0 0 0 0.2 777.6 334.0 1518.3 #15-02-3 15 200 0.24 3 0 0 0 0.2 1094.5 345.1 2130.4 #15-04-1 15 200 0.24 1 0 0 0 0.4 437.1 320.4 812.9 #15-04-2 15 200 0.24 2 0 0 0 0.4 839.9 375.8 1524.6 #15-04-3 15 200 0.24 3 0 0 0 0.4 1214.5 364.0 2118.4 #15-06-1 15 200 0.24 1 0 0 0 0.6 368.7 272.8 818.2 #15-06-2 15 200 0.24 2 0 0 0 0.6 846.2 293.7 1600.1 #15-06-3 15 200 0.24 3 0 0 0 0.6 1312.8 307.0 2237.5 #5-00-1 5 200 0.1 0.5 -45 5 0 0 83.8 124.0 205.0 #5-00-2 5 200 0.1 1 -45 5 0 0 195.3 224.8 397.1 #5-00-3 5 200 0.1 1.5 -45 5 0 0 295.6 314.4 589.8 #5-02-1 5 200 0.1 0.5 -45 5 0 0.2 121.2 214.8 268.0 #5-02-2 5 200 0.1 1 -45 5 0 0.2 265.5 341.2 480.2 #5-02-3 5 200 0.1 1.5 -45 5 0 0.2 396.1 475.5 703.7 #5-04-1 5 200 0.1 0.5 -45 5 0 0.4 82.1 109.0 197.0 #5-04-2 5 200 0.1 1 -45 5 0 0.4 174.9 194.5 373.0 #5-04-3 5 200 0.1 1.5 -45 5 0 0.4 252.3 259.5 527.4 #5-06-1 5 200 0.1 0.5 -45 5 0 0.6 84.2 120.5 202.5 #5-06-2 5 200 0.1 1 -45 5 0 0.6 174.6 198.1 369.8 #5-06-3 5 200 0.1 1.5 -45 5 0 0.6 266.9 281.0 555.1 #6-00-1 6 200 0.1 0.5 -45 5 5 0 61.0 92.9 178.3 #6-00-2 6 200 0.1 1 -45 5 5 0 150.3 179.7 358.1 #6-00-3 6 200 0.1 1.5 -45 5 5 0 229.1 241.4 519.1 #6-02-1 6 200 0.1 0.5 -45 5 5 0.2 103.1 152.0 221.9 #6-02-2 6 200 0.1 1 -45 5 5 0.2 208.2 247.6 421.2 #6-02-3 6 200 0.1 1.5 -45 5 5 0.2 303.5 337.4 605.0 #6-04-1 6 200 0.1 0.5 -45 5 5 0.4 89.1 138.5 205.5 #6-04-2 6 200 0.1 1 -45 5 5 0.4 200.4 232.9 409.3 #6-04-3 6 200 0.1 1.5 -45 5 5 0.4 280.3 308.4 574.3 #6-06-1 6 200 0.1 0.5 -45 5 5 0.6 118.6 166.2 243.3 #6-06-2 6 200 0.1 1 -45 5 5 0.6 234.9 257.6 442.6 #6-06-3 6 200 0.1 1.5 -45 5 5 0.6 325.3 366.3 648.5 #7-00-1 7 200 0.1 0.5 -45 5 -5 0 109.4 163.5 233.7 #7-00-2 7 200 0.1 1 -45 5 -5 0 237.0 284.6 443.5 #7-00-3 7 200 0.1 1.5 -45 5 -5 0 349.8 384.8 638.0 #7-02-1 7 200 0.1 0.5 -45 5 -5 0.2 95.2 130.8 209.1 #7-02-2 7 200 0.1 1 -45 5 -5 0.2 192.4 209.5 382.5 #7-02-3 7 200 0.1 1.5 -45 5 -5 0.2 270.4 270.6 504.0 #7-04-1 7 200 0.1 0.5 -45 5 -5 0.4 91.9 131.6 210.9 #7-04-2 7 200 0.1 1 -45 5 -5 0.4 202.9 230.2 395.2 #7-04-3 7 200 0.1 1.5 -45 5 -5 0.4 316.0 335.1 585.6 #7-06-1 7 200 0.1 0.5 -45 5 -5 0.6 112.5 155.3 225.7 #7-06-2 7 200 0.1 1 -45 5 -5 0.6 204.0 227.8 393.6
A19
#7-06-3 7 200 0.1 1.5 -45 5 -5 0.6 319.2 334.3 592.4 #8-00-1 8 200 0.1 0.5 -45 -5 -5 0 94.9 160.0 219.5 #8-00-2 8 200 0.1 1 -45 -5 -5 0 200.6 268.2 416.0 #8-00-3 8 200 0.1 1.5 -45 -5 -5 0 299.2 365.6 593.3 #8-02-1 8 200 0.1 0.5 -45 -5 -5 0.2 122.4 191.0 250.5 #8-02-2 8 200 0.1 1 -45 -5 -5 0.2 236.8 303.9 444.7 #8-02-3 8 200 0.1 1.5 -45 -5 -5 0.2 357.3 432.6 664.3 #8-04-1 8 200 0.1 0.5 -45 -5 -5 0.4 121.8 208.5 256.2 #8-04-2 8 200 0.1 1 -45 -5 -5 0.4 232.1 313.4 453.6 #8-04-3 8 200 0.1 1.5 -45 -5 -5 0.4 353.0 443.0 657.5 #8-06-1 8 200 0.1 0.5 -45 -5 -5 0.6 127.0 188.9 252.1 #8-06-2 8 200 0.1 1 -45 -5 -5 0.6 245.7 296.3 452.2 #8-06-3 8 200 0.1 1.5 -45 -5 -5 0.6 359.9 425.9 663.5
#11-00-1 11 200 0.1 0.5 -45 -5 0 0 61.9 102.1 181.3 #11-00-2 11 200 0.1 1 -45 -5 0 0 150.0 193.2 361.8 #11-00-3 11 200 0.1 1.5 -45 -5 0 0 238.1 285.5 537.9 #11-02-1 11 200 0.1 0.5 -45 -5 0 0.2 86.3 126.1 207.5 #11-02-2 11 200 0.1 1 -45 -5 0 0.2 199.8 258.2 426.2 #11-02-3 11 200 0.1 1.5 -45 -5 0 0.2 285.7 352.9 604.6 #11-04-1 11 200 0.1 0.5 -45 -5 0 0.4 135.9 201.6 263.0 #11-04-2 11 200 0.1 1 -45 -5 0 0.4 271.7 321.6 490.9 #11-04-3 11 200 0.1 1.5 -45 -5 0 0.4 385.2 438.2 699.3 #11-06-1 11 200 0.1 0.5 -45 -5 0 0.6 112.5 162.8 239.5 #11-06-2 11 200 0.1 1 -45 -5 0 0.6 266.4 297.9 473.4 #11-06-3 11 200 0.1 1.5 -45 -5 0 0.6 388.3 427.2 691.2 #12-00-1 12 200 0.1 0.5 -45 -5 5 0 53.5 93.3 177.5 #12-00-2 12 200 0.1 1 -45 -5 5 0 151.6 204.5 376.0 #12-00-3 12 200 0.1 1.5 -45 -5 5 0 214.6 262.1 519.7 #12-02-1 12 200 0.1 0.5 -45 -5 5 0.2 98.4 144.2 223.5 #12-02-2 12 200 0.1 1 -45 -5 5 0.2 177.5 224.6 382.6 #12-02-3 12 200 0.1 1.5 -45 -5 5 0.2 258.5 309.9 558.4 #12-04-1 12 200 0.1 0.5 -45 -5 5 0.4 108.3 153.8 226.7 #12-04-2 12 200 0.1 1 -45 -5 5 0.4 239.8 303.2 454.2 #12-04-3 12 200 0.1 1.5 -45 -5 5 0.4 348.0 419.3 670.6 #12-06-1 12 200 0.1 0.5 -45 -5 5 0.6 105.2 160.8 229.2 #12-06-2 12 200 0.1 1 -45 -5 5 0.6 227.2 265.6 428.1 #12-06-3 12 200 0.1 1.5 -45 -5 5 0.6 299.4 344.8 619.6 #14-00-1 14 100 0.1 1 0 0 -5 0 317.2 170.7 446.1 #14-00-2 14 100 0.1 2 0 0 -5 0 584.1 169.7 807.5 #14-00-3 14 100 0.1 3 0 0 -5 0 830.2 183.7 1136.0 #14-02-1 14 100 0.1 1 0 0 -5 0.2 365.7 177.7 495.0 #14-02-2 14 100 0.1 2 0 0 -5 0.2 657.5 172.4 860.6 #14-02-3 14 100 0.1 3 0 0 -5 0.2 958.7 202.2 1251.1 #14-04-1 14 100 0.1 1 0 0 -5 0.4 380.0 182.7 517.8 #14-04-2 14 100 0.1 2 0 0 -5 0.4 777.7 193.0 973.0 #14-04-3 14 100 0.1 3 0 0 -5 0.4 1115.5 210.1 1402.0 #14-06-1 14 100 0.1 1 0 0 -5 0.6 466.3 198.0 586.5 #14-06-2 14 100 0.1 2 0 0 -5 0.6 801.1 209.2 996.7 #14-06-3 14 100 0.1 3 0 0 -5 0.6 1229.0 215.4 1466.7 #5-00-1 5 200 0.17 0.5 -45 5 0 0 123.1 193.7 308.8 #5-00-2 5 200 0.17 1 -45 5 0 0 279.5 336.8 599.9 #5-00-3 5 200 0.17 1.5 -45 5 0 0 406.7 430.8 855.4 #5-02-1 5 200 0.17 0.5 -45 5 0 0.2 151.5 277.3 363.9 #5-02-2 5 200 0.17 1 -45 5 0 0.2 328.4 440.8 682.1 #5-02-3 5 200 0.17 1.5 -45 5 0 0.2 498.4 596.6 997.6 #5-04-1 5 200 0.17 0.5 -45 5 0 0.4 111.3 165.3 300.5
A20
#5-04-2 5 200 0.17 1 -45 5 0 0.4 240.1 271.2 557.4 #5-04-3 5 200 0.17 1.5 -45 5 0 0.4 368.9 392.9 825.9 #5-06-1 5 200 0.17 0.5 -45 5 0 0.6 114.0 178.2 300.7 #5-06-2 5 200 0.17 1 -45 5 0 0.6 245.8 275.6 549.1 #5-06-3 5 200 0.17 1.5 -45 5 0 0.6 371.7 390.3 815.8 #6-00-1 6 200 0.17 0.5 -45 5 5 0 100.6 164.1 284.8 #6-00-2 6 200 0.17 1 -45 5 5 0 216.7 255.9 530.8 #6-00-3 6 200 0.17 1.5 -45 5 5 0 337.0 360.9 796.1 #6-02-1 6 200 0.17 0.5 -45 5 5 0.2 125.7 212.1 332.4 #6-02-2 6 200 0.17 1 -45 5 5 0.2 261.2 323.9 600.8 #6-02-3 6 200 0.17 1.5 -45 5 5 0.2 380.0 420.7 853.7 #6-04-1 6 200 0.17 0.5 -45 5 5 0.4 109.7 195.9 309.5 #6-04-2 6 200 0.17 1 -45 5 5 0.4 245.7 309.6 591.1 #6-04-3 6 200 0.17 1.5 -45 5 5 0.4 360.3 416.2 850.3 #6-06-1 6 200 0.17 0.5 -45 5 5 0.6 141.4 232.1 341.8 #6-06-2 6 200 0.17 1 -45 5 5 0.6 282.5 348.4 619.9 #6-06-3 6 200 0.17 1.5 -45 5 5 0.6 393.6 448.3 887.3 #7-00-1 7 200 0.17 0.5 -45 5 -5 0 151.1 248.8 348.7 #7-00-2 7 200 0.17 1 -45 5 -5 0 325.7 392.8 644.2 #7-00-3 7 200 0.17 1.5 -45 5 -5 0 496.4 548.6 936.4 #7-02-1 7 200 0.17 0.5 -45 5 -5 0.2 129.7 195.4 311.9 #7-02-2 7 200 0.17 1 -45 5 -5 0.2 272.5 298.1 559.9 #7-02-3 7 200 0.17 1.5 -45 5 -5 0.2 434.5 444.2 838.6 #7-04-1 7 200 0.17 0.5 -45 5 -5 0.4 125.4 207.6 318.4 #7-04-2 7 200 0.17 1 -45 5 -5 0.4 285.0 334.4 584.2 #7-04-3 7 200 0.17 1.5 -45 5 -5 0.4 449.7 492.9 884.2 #7-06-1 7 200 0.17 0.5 -45 5 -5 0.6 142.3 221.1 331.2 #7-06-2 7 200 0.17 1 -45 5 -5 0.6 281.3 336.2 594.0 #7-06-3 7 200 0.17 1.5 -45 5 -5 0.6 443.7 477.9 873.2 #8-00-1 8 200 0.17 0.5 -45 -5 -5 0 129.7 242.7 327.7 #8-00-2 8 200 0.17 1 -45 -5 -5 0 280.9 385.8 605.6 #8-00-3 8 200 0.17 1.5 -45 -5 -5 0 425.4 560.5 892.4 #8-02-1 8 200 0.17 0.5 -45 -5 -5 0.2 154.3 271.1 356.0 #8-02-2 8 200 0.17 1 -45 -5 -5 0.2 303.6 425.0 649.8 #8-02-3 8 200 0.17 1.5 -45 -5 -5 0.2 453.6 580.9 936.0 #8-04-1 8 200 0.17 0.5 -45 -5 -5 0.4 157.4 291.6 373.7 #8-04-2 8 200 0.17 1 -45 -5 -5 0.4 304.1 443.3 666.7 #8-04-3 8 200 0.17 1.5 -45 -5 -5 0.4 466.2 602.0 937.6 #8-06-1 8 200 0.17 0.5 -45 -5 -5 0.6 158.1 269.0 360.0 #8-06-2 8 200 0.17 1 -45 -5 -5 0.6 330.0 418.9 651.8 #8-06-3 8 200 0.17 1.5 -45 -5 -5 0.6 457.6 591.1 972.6
#11-00-1 11 200 0.17 0.5 -45 -5 0 0 91.4 172.6 293.5 #11-00-2 11 200 0.17 1 -45 -5 0 0 245.3 328.2 588.0 #11-00-3 11 200 0.17 1.5 -45 -5 0 0 386.8 482.5 863.8 #11-02-1 11 200 0.17 0.5 -45 -5 0 0.2 130.5 226.4 338.9 #11-02-2 11 200 0.17 1 -45 -5 0 0.2 255.5 358.1 610.0 #11-02-3 11 200 0.17 1.5 -45 -5 0 0.2 357.7 467.2 840.2 #11-04-1 11 200 0.17 0.5 -45 -5 0 0.4 144.1 256.4 347.1 #11-04-2 11 200 0.17 1 -45 -5 0 0.4 306.4 400.0 662.8 #11-04-3 11 200 0.17 1.5 -45 -5 0 0.4 423.2 524.2 930.1 #11-06-1 11 200 0.17 0.5 -45 -5 0 0.6 135.8 228.8 327.0 #11-06-2 11 200 0.17 1 -45 -5 0 0.6 304.9 381.5 619.2 #11-06-3 11 200 0.17 1.5 -45 -5 0 0.6 446.5 508.8 930.5 #12-00-1 12 200 0.17 0.5 -45 -5 5 0 79.7 152.0 277.3 #12-00-2 12 200 0.17 1 -45 -5 5 0 227.7 303.2 567.8 #12-00-3 12 200 0.17 1.5 -45 -5 5 0 299.9 381.0 785.9
A21
#12-02-1 12 200 0.17 0.5 -45 -5 5 0.2 112.6 201.0 324.9 #12-02-2 12 200 0.17 1 -45 -5 5 0.2 218.2 318.1 577.5 #12-02-3 12 200 0.17 1.5 -45 -5 5 0.2 303.3 398.7 808.4 #12-04-1 12 200 0.17 0.5 -45 -5 5 0.4 132.5 232.8 356.9 #12-04-2 12 200 0.17 1 -45 -5 5 0.4 277.5 363.3 637.8 #12-04-3 12 200 0.17 1.5 -45 -5 5 0.4 403.7 497.8 910.4 #12-06-1 12 200 0.17 0.5 -45 -5 5 0.6 128.6 207.1 336.4 #12-06-2 12 200 0.17 1 -45 -5 5 0.6 245.7 322.9 598.7 #12-06-3 12 200 0.17 1.5 -45 -5 5 0.6 335.6 402.2 832.9 #14-00-1 14 100 0.17 1 0 0 -5 0 438.4 259.7 682.9 #14-00-2 14 100 0.17 2 0 0 -5 0 973.3 319.1 1304.5 #14-00-3 14 100 0.17 3 0 0 -5 0 1380.3 336.9 1840.3 #14-02-1 14 100 0.17 1 0 0 -5 0.2 473.6 272.0 744.4 #14-02-2 14 100 0.17 2 0 0 -5 0.2 981.9 308.1 1334.7 #14-02-3 14 100 0.17 3 0 0 -5 0.2 1506.0 354.4 1966.6 #14-04-1 14 100 0.17 1 0 0 -5 0.4 507.7 282.5 771.6 #14-04-2 14 100 0.17 2 0 0 -5 0.4 1103.8 315.6 1455.0 #14-04-3 14 100 0.17 3 0 0 -5 0.4 1606.0 336.1 2079.1 #14-06-1 14 100 0.17 1 0 0 -5 0.6 615.2 298.9 825.8 #14-06-2 14 100 0.17 2 0 0 -5 0.6 1207.3 338.8 1520.0 #14-06-3 14 100 0.17 3 0 0 -5 0.6 1673.7 357.6 2122.6 #5-00-1 5 200 0.24 0.5 -45 5 0 0 140.3 251.3 404.0 #5-00-2 5 200 0.24 1 -45 5 0 0 338.7 431.4 781.2 #5-00-3 5 200 0.24 1.5 -45 5 0 0 567.3 624.9 1170.8 #5-02-1 5 200 0.24 0.5 -45 5 0 0.2 156.3 341.1 464.0 #5-02-2 5 200 0.24 1 -45 5 0 0.2 372.7 528.4 857.4 #5-02-3 5 200 0.24 1.5 -45 5 0 0.2 578.0 707.7 1240.6 #5-04-1 5 200 0.24 0.5 -45 5 0 0.4 124.9 207.8 389.0 #5-04-2 5 200 0.24 1 -45 5 0 0.4 288.3 363.3 737.2 #5-04-3 5 200 0.24 1.5 -45 5 0 0.4 459.7 513.2 1079.0 #5-06-1 5 200 0.24 0.5 -45 5 0 0.6 128.3 219.4 382.2 #5-06-2 5 200 0.24 1 -45 5 0 0.6 302.3 367.1 722.5 #5-06-3 5 200 0.24 1.5 -45 5 0 0.6 486.5 521.9 1077.6 #6-00-1 6 200 0.24 0.5 -45 5 5 0 113.1 206.9 363.0 #6-00-2 6 200 0.24 1 -45 5 5 0 273.6 336.7 706.1 #6-00-3 6 200 0.24 1.5 -45 5 5 0 426.3 469.7 1054.0 #6-02-1 6 200 0.24 0.5 -45 5 5 0.2 138.0 257.8 426.1 #6-02-2 6 200 0.24 1 -45 5 5 0.2 300.0 387.6 763.4 #6-02-3 6 200 0.24 1.5 -45 5 5 0.2 449.9 509.0 1102.2 #6-04-1 6 200 0.24 0.5 -45 5 5 0.4 122.9 240.1 399.3 #6-04-2 6 200 0.24 1 -45 5 5 0.4 279.2 375.2 759.7 #6-04-3 6 200 0.24 1.5 -45 5 5 0.4 431.7 497.6 1091.6 #6-06-1 6 200 0.24 0.5 -45 5 5 0.6 148.6 277.5 428.5 #6-06-2 6 200 0.24 1 -45 5 5 0.6 306.3 403.8 784.2 #6-06-3 6 200 0.24 1.5 -45 5 5 0.6 459.1 538.9 1122.1 #7-00-1 7 200 0.24 0.5 -45 5 -5 0 169.7 300.6 438.6 #7-00-2 7 200 0.24 1 -45 5 -5 0 410.9 531.6 846.6 #7-00-3 7 200 0.24 1.5 -45 5 -5 0 672.5 771.9 1260.9 #7-02-1 7 200 0.24 0.5 -45 5 -5 0.2 140.9 244.7 401.2 #7-02-2 7 200 0.24 1 -45 5 -5 0.2 338.9 389.4 730.6 #7-02-3 7 200 0.24 1.5 -45 5 -5 0.2 591.7 666.3 1174.8 #7-04-1 7 200 0.24 0.5 -45 5 -5 0.4 140.3 268.6 411.9 #7-04-2 7 200 0.24 1 -45 5 -5 0.4 351.8 448.2 774.6 #7-04-3 7 200 0.24 1.5 -45 5 -5 0.4 597.3 655.3 1152.0 #7-06-1 7 200 0.24 0.5 -45 5 -5 0.6 155.2 263.1 412.5 #7-06-2 7 200 0.24 1 -45 5 -5 0.6 360.2 447.7 774.0
A22
#7-06-3 7 200 0.24 1.5 -45 5 -5 0.6 577.5 634.2 1148.9 #8-00-1 8 200 0.24 0.5 -45 -5 -5 0 144.3 311.5 420.4 #8-00-2 8 200 0.24 1 -45 -5 -5 0 354.0 518.4 791.5 #8-00-3 8 200 0.24 1.5 -45 -5 -5 0 581.6 767.3 1193.1 #8-02-1 8 200 0.24 0.5 -45 -5 -5 0.2 167.8 353.7 465.6 #8-02-2 8 200 0.24 1 -45 -5 -5 0.2 366.3 541.4 834.6 #8-02-3 8 200 0.24 1.5 -45 -5 -5 0.2 591.0 772.3 1235.0 #8-04-1 8 200 0.24 0.5 -45 -5 -5 0.4 168.1 364.6 472.2 #8-04-2 8 200 0.24 1 -45 -5 -5 0.4 363.4 565.5 843.9 #8-04-3 8 200 0.24 1.5 -45 -5 -5 0.4 610.8 830.3 1250.2 #8-06-1 8 200 0.24 0.5 -45 -5 -5 0.6 167.6 341.1 453.3 #8-06-2 8 200 0.24 1 -45 -5 -5 0.6 374.2 542.8 839.6 #8-06-3 8 200 0.24 1.5 -45 -5 -5 0.6 559.1 750.4 1207.7
#11-00-1 11 200 0.24 0.5 -45 -5 0 0 116.7 242.6 395.1 #11-00-2 11 200 0.24 1 -45 -5 0 0 284.3 418.6 746.1 #11-00-3 11 200 0.24 1.5 -45 -5 0 0 527.3 651.0 1159.9 #11-02-1 11 200 0.24 0.5 -45 -5 0 0.2 151.4 269.9 423.7 #11-02-2 11 200 0.24 1 -45 -5 0 0.2 297.8 436.1 761.6 #11-02-3 11 200 0.24 1.5 -45 -5 0 0.2 436.5 607.6 1096.0 #11-04-1 11 200 0.24 0.5 -45 -5 0 0.4 143.8 295.9 430.3 #11-04-2 11 200 0.24 1 -45 -5 0 0.4 345.9 497.2 846.0 #11-04-3 11 200 0.24 1.5 -45 -5 0 0.4 536.6 668.9 1223.3 #11-06-1 11 200 0.24 0.5 -45 -5 0 0.6 131.8 282.9 426.7 #11-06-2 11 200 0.24 1 -45 -5 0 0.6 349.5 462.3 802.3 #11-06-3 11 200 0.24 1.5 -45 -5 0 0.6 564.6 682.2 1224.4 #12-00-1 12 200 0.24 0.5 -45 -5 5 0 92.6 204.7 365.4 #12-00-2 12 200 0.24 1 -45 -5 5 0 268.3 385.4 740.9 #12-00-3 12 200 0.24 1.5 -45 -5 5 0 428.7 534.7 1083.6 #12-02-1 12 200 0.24 0.5 -45 -5 5 0.2 116.9 247.2 408.1 #12-02-2 12 200 0.24 1 -45 -5 5 0.2 237.8 367.0 737.5 #12-02-3 12 200 0.24 1.5 -45 -5 5 0.2 374.3 498.2 1037.4 #12-04-1 12 200 0.24 0.5 -45 -5 5 0.4 139.3 272.3 436.9 #12-04-2 12 200 0.24 1 -45 -5 5 0.4 322.6 471.6 821.2 #12-04-3 12 200 0.24 1.5 -45 -5 5 0.4 522.2 655.8 1205.8 #12-06-1 12 200 0.24 0.5 -45 -5 5 0.6 130.5 251.9 422.5 #12-06-2 12 200 0.24 1 -45 -5 5 0.6 253.0 361.9 735.1 #12-06-3 12 200 0.24 1.5 -45 -5 5 0.6 387.6 504.4 1082.6 #14-00-1 14 100 0.24 1 0 0 -5 0 514.4 359.2 896.1 #14-00-2 14 100 0.24 2 0 0 -5 0 1226.9 442.6 1748.8 #14-00-3 14 100 0.24 3 0 0 -5 0 1770.5 501.6 2472.5 #14-02-1 14 100 0.24 1 0 0 -5 0.2 561.7 372.0 958.9 #14-02-2 14 100 0.24 2 0 0 -5 0.2 1249.0 434.7 1771.5 #14-02-3 14 100 0.24 3 0 0 -5 0.2 1831.8 480.1 2590.1 #14-04-1 14 100 0.24 1 0 0 -5 0.4 579.9 370.7 997.5 #14-04-2 14 100 0.24 2 0 0 -5 0.4 1315.3 414.7 1886.6 #14-04-3 14 100 0.24 3 0 0 -5 0.4 1867.2 432.2 2625.6 #14-06-1 14 100 0.24 1 0 0 -5 0.6 665.8 386.1 1033.2 #14-06-2 14 100 0.24 2 0 0 -5 0.6 1419.8 449.9 1959.8 #14-06-3 14 100 0.24 3 0 0 -5 0.6 1952.5 459.6 2697.7 #5-00-1 5 300 0.1 0.5 -45 5 0 0 84.4 115.9 188.1 #5-00-2 5 300 0.1 1 -45 5 0 0 193.5 214.2 369.3 #5-00-3 5 300 0.1 1.5 -45 5 0 0 308.6 318.8 563.3 #5-02-1 5 300 0.1 0.5 -45 5 0 0.2 147.1 270.4 276.9 #5-02-2 5 300 0.1 1 -45 5 0 0.2 320.3 437.8 493.9 #5-02-3 5 300 0.1 1.5 -45 5 0 0.2 477.0 588.8 716.9 #5-04-1 5 300 0.1 0.5 -45 5 0 0.4 92.3 119.8 196.6
A23
#5-04-2 5 300 0.1 1 -45 5 0 0.4 182.9 200.0 362.6 #5-04-3 5 300 0.1 1.5 -45 5 0 0.4 279.3 287.3 538.1 #5-06-1 5 300 0.1 0.5 -45 5 0 0.6 78.7 107.2 184.3 #5-06-2 5 300 0.1 1 -45 5 0 0.6 178.5 193.5 352.9 #5-06-3 5 300 0.1 1.5 -45 5 0 0.6 271.2 275.6 519.6 #6-00-1 6 300 0.1 0.5 -45 5 5 0 69.5 99.1 173.8 #6-00-2 6 300 0.1 1 -45 5 5 0 156.1 180.1 347.8 #6-00-3 6 300 0.1 1.5 -45 5 5 0 241.7 253.0 511.2 #6-02-1 6 300 0.1 0.5 -45 5 5 0.2 100.4 159.0 232.1 #6-02-2 6 300 0.1 1 -45 5 5 0.2 209.6 251.3 416.6 #6-02-3 6 300 0.1 1.5 -45 5 5 0.2 303.5 332.3 597.7 #6-04-1 6 300 0.1 0.5 -45 5 5 0.4 96.4 150.9 213.3 #6-04-2 6 300 0.1 1 -45 5 5 0.4 199.7 238.3 403.5 #6-04-3 6 300 0.1 1.5 -45 5 5 0.4 292.8 315.5 575.2 #6-06-1 6 300 0.1 0.5 -45 5 5 0.6 115.4 160.6 232.8 #6-06-2 6 300 0.1 1 -45 5 5 0.6 215.4 249.6 421.6 #6-06-3 6 300 0.1 1.5 -45 5 5 0.6 312.1 330.1 592.6 #7-00-1 7 300 0.1 0.5 -45 5 -5 0 101.5 143.2 206.3 #7-00-2 7 300 0.1 1 -45 5 -5 0 243.9 280.3 415.7 #7-00-3 7 300 0.1 1.5 -45 5 -5 0 354.9 371.6 592.9 #7-02-1 7 300 0.1 0.5 -45 5 -5 0.2 86.9 110.6 185.3 #7-02-2 7 300 0.1 1 -45 5 -5 0.2 202.5 213.7 365.4 #7-02-3 7 300 0.1 1.5 -45 5 -5 0.2 306.6 316.9 547.0 #7-04-1 7 300 0.1 0.5 -45 5 -5 0.4 91.0 123.9 189.2 #7-04-2 7 300 0.1 1 -45 5 -5 0.4 207.8 222.1 364.0 #7-04-3 7 300 0.1 1.5 -45 5 -5 0.4 365.6 390.9 655.9 #7-06-1 7 300 0.1 0.5 -45 5 -5 0.6 96.6 125.8 196.1 #7-06-2 7 300 0.1 1 -45 5 -5 0.6 209.3 220.6 367.2 #7-06-3 7 300 0.1 1.5 -45 5 -5 0.6 328.4 331.7 559.1 #8-00-1 8 300 0.1 0.5 -45 -5 -5 0 88.1 146.9 196.2 #8-00-2 8 300 0.1 1 -45 -5 -5 0 209.3 277.9 394.2 #8-00-3 8 300 0.1 1.5 -45 -5 -5 0 311.6 386.8 571.9 #8-02-1 8 300 0.1 0.5 -45 -5 -5 0.2 111.9 181.0 233.9 #8-02-2 8 300 0.1 1 -45 -5 -5 0.2 235.5 310.3 444.1 #8-02-3 8 300 0.1 1.5 -45 -5 -5 0.2 351.9 441.1 649.3 #8-04-1 8 300 0.1 0.5 -45 -5 -5 0.4 109.4 188.1 236.7 #8-04-2 8 300 0.1 1 -45 -5 -5 0.4 237.0 326.2 445.1 #8-04-3 8 300 0.1 1.5 -45 -5 -5 0.4 364.2 465.0 659.6 #8-06-1 8 300 0.1 0.5 -45 -5 -5 0.6 117.4 186.3 238.2 #8-06-2 8 300 0.1 1 -45 -5 -5 0.6 235.5 304.5 439.7 #8-06-3 8 300 0.1 1.5 -45 -5 -5 0.6 344.5 420.2 635.6
#11-00-1 11 300 0.1 0.5 -45 -5 0 0 73.8 117.4 185.8 #11-00-2 11 300 0.1 1 -45 -5 0 0 152.6 197.5 350.9 #11-00-3 11 300 0.1 1.5 -45 -5 0 0 236.3 288.8 521.3 #11-02-1 11 300 0.1 0.5 -45 -5 0 0.2 105.5 164.3 235.9 #11-02-2 11 300 0.1 1 -45 -5 0 0.2 189.3 246.7 395.5 #11-02-3 11 300 0.1 1.5 -45 -5 0 0.2 229.5 291.5 477.2 #11-04-1 11 300 0.1 0.5 -45 -5 0 0.4 114.5 179.3 252.0 #11-04-2 11 300 0.1 1 -45 -5 0 0.4 249.0 319.5 494.0 #11-04-3 11 300 0.1 1.5 -45 -5 0 0.4 338.9 401.2 679.9 #11-06-1 11 300 0.1 0.5 -45 -5 0 0.6 105.5 167.8 239.0 #11-06-2 11 300 0.1 1 -45 -5 0 0.6 227.7 269.5 444.4 #11-06-3 11 300 0.1 1.5 -45 -5 0 0.6 327.0 364.3 652.6 #12-00-1 12 300 0.1 0.5 -45 -5 5 0 64.6 109.9 182.2 #12-00-2 12 300 0.1 1 -45 -5 5 0 137.5 176.5 340.0 #12-00-3 12 300 0.1 1.5 -45 -5 5 0 226.9 280.4 526.0
A24
#12-02-1 12 300 0.1 0.5 -45 -5 5 0.2 91.8 152.2 237.1 #12-02-2 12 300 0.1 1 -45 -5 5 0.2 166.7 229.2 403.7 #12-02-3 12 300 0.1 1.5 -45 -5 5 0.2 226.9 297.6 554.1 #12-04-1 12 300 0.1 0.5 -45 -5 5 0.4 138.3 196.3 252.5 #12-04-2 12 300 0.1 1 -45 -5 5 0.4 307.3 370.1 474.4 #12-04-3 12 300 0.1 1.5 -45 -5 5 0.4 443.3 505.2 679.8 #12-06-1 12 300 0.1 0.5 -45 -5 5 0.6 92.9 153.3 237.3 #12-06-2 12 300 0.1 1 -45 -5 5 0.6 181.9 235.8 421.0 #12-06-3 12 300 0.1 1.5 -45 -5 5 0.6 240.5 291.0 571.3 #14-00-1 14 200 0.1 1 0 0 -5 0 304.9 163.6 421.7 #14-00-2 14 200 0.1 2 0 0 -5 0 630.6 196.5 804.8 #14-00-3 14 200 0.1 3 0 0 -5 0 887.2 206.3 1136.0 #14-02-1 14 200 0.1 1 0 0 -5 0.2 332.8 173.8 460.0 #14-02-2 14 200 0.1 2 0 0 -5 0.2 664.1 186.0 860.7 #14-02-3 14 200 0.1 3 0 0 -5 0.2 940.6 199.6 1241.9 #14-04-1 14 200 0.1 1 0 0 -5 0.4 327.0 157.5 474.0 #14-04-2 14 200 0.1 2 0 0 -5 0.4 652.6 178.9 902.1 #14-04-3 14 200 0.1 3 0 0 -5 0.4 955.8 193.3 1298.2 #14-06-1 14 200 0.1 1 0 0 -5 0.6 386.8 177.0 510.0 #14-06-2 14 200 0.1 2 0 0 -5 0.6 725.8 200.1 958.9 #14-06-3 14 200 0.1 3 0 0 -5 0.6 1009.8 212.7 1346.6 #5-00-1 5 300 0.17 0.5 -45 5 0 0 125.1 194.0 299.7 #5-00-2 5 300 0.17 1 -45 5 0 0 285.7 328.3 572.3 #5-00-3 5 300 0.17 1.5 -45 5 0 0 442.8 461.6 856.5 #5-02-1 5 300 0.17 0.5 -45 5 0 0.2 173.3 342.7 370.4 #5-02-2 5 300 0.17 1 -45 5 0 0.2 382.1 542.7 674.3 #5-02-3 5 300 0.17 1.5 -45 5 0 0.2 580.7 723.8 989.6 #5-04-1 5 300 0.17 0.5 -45 5 0 0.4 122.2 178.0 296.4 #5-04-2 5 300 0.17 1 -45 5 0 0.4 265.5 298.7 558.7 #5-04-3 5 300 0.17 1.5 -45 5 0 0.4 401.4 419.3 827.4 #5-06-1 5 300 0.17 0.5 -45 5 0 0.6 122.4 177.3 291.0 #5-06-2 5 300 0.17 1 -45 5 0 0.6 270.9 302.7 547.6 #5-06-3 5 300 0.17 1.5 -45 5 0 0.6 404.3 422.4 816.2 #6-00-1 6 300 0.17 0.5 -45 5 5 0 99.9 158.0 269.8 #6-00-2 6 300 0.17 1 -45 5 5 0 244.3 290.6 550.1 #6-00-3 6 300 0.17 1.5 -45 5 5 0 361.8 386.2 800.9 #6-02-1 6 300 0.17 0.5 -45 5 5 0.2 126.0 204.3 329.5 #6-02-2 6 300 0.17 1 -45 5 5 0.2 261.3 332.2 611.4 #6-02-3 6 300 0.17 1.5 -45 5 5 0.2 386.5 439.2 870.1 #6-04-1 6 300 0.17 0.5 -45 5 5 0.4 111.8 190.0 307.4 #6-04-2 6 300 0.17 1 -45 5 5 0.4 253.8 308.5 583.5 #6-04-3 6 300 0.17 1.5 -45 5 5 0.4 381.2 415.7 837.8 #6-06-1 6 300 0.17 0.5 -45 5 5 0.6 129.9 203.9 320.5 #6-06-2 6 300 0.17 1 -45 5 5 0.6 279.9 333.0 599.8 #6-06-3 6 300 0.17 1.5 -45 5 5 0.6 396.6 426.7 855.3 #7-00-1 7 300 0.17 0.5 -45 5 -5 0 151.5 231.8 323.5 #7-00-2 7 300 0.17 1 -45 5 -5 0 333.9 400.5 624.0 #7-00-3 7 300 0.17 1.5 -45 5 -5 0 487.1 521.5 886.0 #7-02-1 7 300 0.17 0.5 -45 5 -5 0.2 129.8 184.0 293.2 #7-02-2 7 300 0.17 1 -45 5 -5 0.2 287.1 320.9 562.3 #7-02-3 7 300 0.17 1.5 -45 5 -5 0.2 457.1 503.6 874.6 #7-04-1 7 300 0.17 0.5 -45 5 -5 0.4 131.2 195.5 295.3 #7-04-2 7 300 0.17 1 -45 5 -5 0.4 296.1 334.0 560.0 #7-04-3 7 300 0.17 1.5 -45 5 -5 0.4 464.8 493.9 847.2 #7-06-1 7 300 0.17 0.5 -45 5 -5 0.6 137.6 201.7 305.5 #7-06-2 7 300 0.17 1 -45 5 -5 0.6 299.6 334.4 570.7
A25
#7-06-3 7 300 0.17 1.5 -45 5 -5 0.6 460.7 477.0 852.2 #8-00-1 8 300 0.17 0.5 -45 -5 -5 0 130.9 238.6 308.4 #8-00-2 8 300 0.17 1 -45 -5 -5 0 295.4 413.1 595.2 #8-00-3 8 300 0.17 1.5 -45 -5 -5 0 427.6 555.1 856.6 #8-02-1 8 300 0.17 0.5 -45 -5 -5 0.2 146.8 263.5 347.4 #8-02-2 8 300 0.17 1 -45 -5 -5 0.2 313.3 428.8 642.3 #8-02-3 8 300 0.17 1.5 -45 -5 -5 0.2 464.8 594.2 942.6 #8-04-1 8 300 0.17 0.5 -45 -5 -5 0.4 143.5 270.7 347.1 #8-04-2 8 300 0.17 1 -45 -5 -5 0.4 304.9 441.3 640.0 #8-04-3 8 300 0.17 1.5 -45 -5 -5 0.4 453.8 606.3 931.0 #8-06-1 8 300 0.17 0.5 -45 -5 -5 0.6 141.4 251.2 335.6 #8-06-2 8 300 0.17 1 -45 -5 -5 0.6 301.2 411.8 625.1 #8-06-3 8 300 0.17 1.5 -45 -5 -5 0.6 431.8 552.8 907.5
#11-00-1 11 300 0.17 0.5 -45 -5 0 0 104.4 186.4 289.5 #11-00-2 11 300 0.17 1 -45 -5 0 0 221.0 302.2 542.8 #11-00-3 11 300 0.17 1.5 -45 -5 0 0 327.9 411.1 784.0 #11-02-1 11 300 0.17 0.5 -45 -5 0 0.2 140.7 235.3 353.8 #11-02-2 11 300 0.17 1 -45 -5 0 0.2 235.3 327.4 565.0 #11-02-3 11 300 0.17 1.5 -45 -5 0 0.2 285.4 384.4 690.6 #11-04-1 11 300 0.17 0.5 -45 -5 0 0.4 135.8 231.0 349.0 #11-04-2 11 300 0.17 1 -45 -5 0 0.4 309.6 405.3 670.0 #11-04-3 11 300 0.17 1.5 -45 -5 0 0.4 404.5 498.0 937.0 #11-06-1 11 300 0.17 0.5 -45 -5 0 0.6 122.8 217.7 323.5 #11-06-2 11 300 0.17 1 -45 -5 0 0.6 272.4 354.6 620.3 #11-06-3 11 300 0.17 1.5 -45 -5 0 0.6 365.2 441.6 884.1 #12-00-1 12 300 0.17 0.5 -45 -5 5 0 89.5 167.5 281.8 #12-00-2 12 300 0.17 1 -45 -5 5 0 197.0 299.0 547.9 #12-00-3 12 300 0.17 1.5 -45 -5 5 0 299.5 369.3 771.9 #12-02-1 12 300 0.17 0.5 -45 -5 5 0.2 103.7 195.2 326.0 #12-02-2 12 300 0.17 1 -45 -5 5 0.2 216.5 321.0 585.1 #12-02-3 12 300 0.17 1.5 -45 -5 5 0.2 283.0 376.1 790.4 #12-04-1 12 300 0.17 0.5 -45 -5 5 0.4 182.3 283.7 373.4 #12-04-2 12 300 0.17 1 -45 -5 5 0.4 382.5 480.7 667.4 #12-04-3 12 300 0.17 1.5 -45 -5 5 0.4 496.4 585.1 915.6 #12-06-1 12 300 0.17 0.5 -45 -5 5 0.6 104.7 189.4 318.1 #12-06-2 12 300 0.17 1 -45 -5 5 0.6 225.6 301.2 582.7 #12-06-3 12 300 0.17 1.5 -45 -5 5 0.6 291.8 373.0 800.7 #14-00-1 14 200 0.17 1 0 0 -5 0 395.1 250.4 640.8 #14-00-2 14 200 0.17 2 0 0 -5 0 971.5 346.8 1321.5 #14-00-3 14 200 0.17 3 0 0 -5 0 1227.0 325.5 1737.1 #14-02-1 14 200 0.17 1 0 0 -5 0.2 415.7 236.3 691.3 #14-02-2 14 200 0.17 2 0 0 -5 0.2 887.3 311.5 1283.6 #14-02-3 14 200 0.17 3 0 0 -5 0.2 1239.2 295.4 1830.5 #14-04-1 14 200 0.17 1 0 0 -5 0.4 399.7 220.2 691.4 #14-04-2 14 200 0.17 2 0 0 -5 0.4 842.4 276.1 1283.5 #14-04-3 14 200 0.17 3 0 0 -5 0.4 1249.3 310.7 1950.2 #14-06-1 14 200 0.17 1 0 0 -5 0.6 453.4 242.6 736.5 #14-06-2 14 200 0.17 2 0 0 -5 0.6 915.7 307.4 1364.2 #14-06-3 14 200 0.17 3 0 0 -5 0.6 1312.0 321.6 1913.9 #5-00-1 5 300 0.24 0.5 -45 5 0 0 147.0 257.6 400.4 #5-00-2 5 300 0.24 1 -45 5 0 0 342.8 408.0 750.5 #5-00-3 5 300 0.24 1.5 -45 5 0 0 520.7 564.0 1114.3 #5-02-1 5 300 0.24 0.5 -45 5 0 0.2 179.0 399.4 452.6 #5-02-2 5 300 0.24 1 -45 5 0 0.2 427.2 631.3 848.8 #5-02-3 5 300 0.24 1.5 -45 5 0 0.2 658.8 844.9 1253.1 #5-04-1 5 300 0.24 0.5 -45 5 0 0.4 144.3 224.7 387.2
A26
#5-04-2 5 300 0.24 1 -45 5 0 0.4 321.4 375.7 737.4 #5-04-3 5 300 0.24 1.5 -45 5 0 0.4 480.0 522.2 1082.0 #5-06-1 5 300 0.24 0.5 -45 5 0 0.6 143.2 247.4 393.6 #5-06-2 5 300 0.24 1 -45 5 0 0.6 325.2 385.2 723.2 #5-06-3 5 300 0.24 1.5 -45 5 0 0.6 487.3 522.9 1072.2 #6-00-1 6 300 0.24 0.5 -45 5 5 0 119.8 214.2 361.3 #6-00-2 6 300 0.24 1 -45 5 5 0 296.5 362.4 722.4 #6-00-3 6 300 0.24 1.5 -45 5 5 0 441.0 487.8 1060.3 #6-02-1 6 300 0.24 0.5 -45 5 5 0.2 137.3 245.4 414.7 #6-02-2 6 300 0.24 1 -45 5 5 0.2 295.2 377.7 758.8 #6-02-3 6 300 0.24 1.5 -45 5 5 0.2 455.9 509.7 1105.9 #6-04-1 6 300 0.24 0.5 -45 5 5 0.4 122.6 230.7 389.9 #6-04-2 6 300 0.24 1 -45 5 5 0.4 294.3 373.0 750.6 #6-04-3 6 300 0.24 1.5 -45 5 5 0.4 438.9 483.4 1071.5 #6-06-1 6 300 0.24 0.5 -45 5 5 0.6 140.1 243.6 409.8 #6-06-2 6 300 0.24 1 -45 5 5 0.6 315.7 383.8 763.2 #6-06-3 6 300 0.24 1.5 -45 5 5 0.6 431.2 479.7 1068.4 #7-00-1 7 300 0.24 0.5 -45 5 -5 0 182.5 329.8 443.9 #7-00-2 7 300 0.24 1 -45 5 -5 0 391.3 474.1 798.4 #7-00-3 7 300 0.24 1.5 -45 5 -5 0 594.8 655.0 1164.8 #7-02-1 7 300 0.24 0.5 -45 5 -5 0.2 150.9 245.0 389.6 #7-02-2 7 300 0.24 1 -45 5 -5 0.2 354.8 433.6 765.7 #7-02-3 7 300 0.24 1.5 -45 5 -5 0.2 531.0 557.7 1082.1 #7-04-1 7 300 0.24 0.5 -45 5 -5 0.4 150.8 258.5 391.6 #7-04-2 7 300 0.24 1 -45 5 -5 0.4 360.9 422.4 741.4 #7-04-3 7 300 0.24 1.5 -45 5 -5 0.4 561.6 606.9 1109.2 #7-06-1 7 300 0.24 0.5 -45 5 -5 0.6 159.0 257.4 394.8 #7-06-2 7 300 0.24 1 -45 5 -5 0.6 358.4 417.7 745.5 #7-06-3 7 300 0.24 1.5 -45 5 -5 0.6 553.5 582.9 1111.8 #8-00-1 8 300 0.24 0.5 -45 -5 -5 0 153.4 316.4 408.0 #8-00-2 8 300 0.24 1 -45 -5 -5 0 346.6 515.4 775.4 #8-00-3 8 300 0.24 1.5 -45 -5 -5 0 512.5 686.6 1115.5 #8-02-1 8 300 0.24 0.5 -45 -5 -5 0.2 169.3 336.0 444.6 #8-02-2 8 300 0.24 1 -45 -5 -5 0.2 364.0 531.1 821.7 #8-02-3 8 300 0.24 1.5 -45 -5 -5 0.2 536.4 716.5 1195.6 #8-04-1 8 300 0.24 0.5 -45 -5 -5 0.4 165.8 344.2 446.6 #8-04-2 8 300 0.24 1 -45 -5 -5 0.4 353.2 534.7 816.1 #8-04-3 8 300 0.24 1.5 -45 -5 -5 0.4 534.8 739.2 1193.1 #8-06-1 8 300 0.24 0.5 -45 -5 -5 0.6 148.8 308.4 415.3 #8-06-2 8 300 0.24 1 -45 -5 -5 0.6 360.4 532.1 837.0 #8-06-3 8 300 0.24 1.5 -45 -5 -5 0.6 510.4 692.3 1183.4
#11-00-1 11 300 0.24 0.5 -45 -5 0 0 148.8 302.8 422.1 #11-00-2 11 300 0.24 1 -45 -5 0 0 315.1 447.9 764.7 #11-00-3 11 300 0.24 1.5 -45 -5 0 0 353.3 474.2 1001.6 #11-02-1 11 300 0.24 0.5 -45 -5 0 0.2 156.1 291.6 453.0 #11-02-2 11 300 0.24 1 -45 -5 0 0.2 276.1 400.8 732.2 #11-02-3 11 300 0.24 1.5 -45 -5 0 0.2 343.0 477.6 894.2 #11-04-1 11 300 0.24 0.5 -45 -5 0 0.4 157.2 298.9 450.3 #11-04-2 11 300 0.24 1 -45 -5 0 0.4 284.9 399.6 781.6 #11-04-3 11 300 0.24 1.5 -45 -5 0 0.4 514.4 644.7 1234.8 #11-06-1 11 300 0.24 0.5 -45 -5 0 0.6 133.9 273.3 413.3 #11-06-2 11 300 0.24 1 -45 -5 0 0.6 309.9 436.5 795.8 #11-06-3 11 300 0.24 1.5 -45 -5 0 0.6 463.3 578.4 1156.2 #12-00-1 12 300 0.24 0.5 -45 -5 5 0 118.2 252.0 387.2 #12-00-2 12 300 0.24 1 -45 -5 5 0 329.5 475.7 785.0 #12-00-3 12 300 0.24 1.5 -45 -5 5 0 451.2 576.2 1107.7
A27
#12-02-1 12 300 0.24 0.5 -45 -5 5 0.2 123.1 260.9 428.7 #12-02-2 12 300 0.24 1 -45 -5 5 0.2 331.4 471.0 818.8 #12-02-3 12 300 0.24 1.5 -45 -5 5 0.2 325.3 442.6 1011.2 #12-04-1 12 300 0.24 0.5 -45 -5 5 0.4 173.7 310.8 445.6 #12-04-2 12 300 0.24 1 -45 -5 5 0.4 395.6 519.0 814.6 #12-04-3 12 300 0.24 1.5 -45 -5 5 0.4 705.4 845.2 1287.5 #12-06-1 12 300 0.24 0.5 -45 -5 5 0.6 110.5 230.2 402.1 #12-06-2 12 300 0.24 1 -45 -5 5 0.6 306.3 414.3 776.0 #12-06-3 12 300 0.24 1.5 -45 -5 5 0.6 491.7 598.7 1126.0 #14-00-1 14 200 0.24 1 0 0 -5 0 426.8 318.6 775.9 #14-00-2 14 200 0.24 2 0 0 -5 0 1110.6 464.7 1675.8 #14-00-3 14 200 0.24 3 0 0 -5 0 1444.8 411.3 2317.4 #14-02-1 14 200 0.24 1 0 0 -5 0.2 443.4 324.7 842.3 #14-02-2 14 200 0.24 2 0 0 -5 0.2 1017.0 391.3 1672.6 #14-02-3 14 200 0.24 3 0 0 -5 0.2 1425.2 392.4 2332.8 #14-04-1 14 200 0.24 1 0 0 -5 0.4 425.8 284.5 859.3 #14-04-2 14 200 0.24 2 0 0 -5 0.4 1035.3 368.6 1759.2 #14-04-3 14 200 0.24 3 0 0 -5 0.4 1480.1 387.0 2447.4 #14-06-1 14 200 0.24 1 0 0 -5 0.6 462.1 305.4 881.0 #14-06-2 14 200 0.24 2 0 0 -5 0.6 1055.6 397.5 1713.0 #14-06-3 14 200 0.24 3 0 0 -5 0.6 1568.0 425.0 2519.8
